{"id":43,"date":"2023-09-17T10:56:20","date_gmt":"2023-09-17T10:56:20","guid":{"rendered":"https:\/\/mathority.org\/id\/maxima-minima-dari-suatu-fungsi-relatif-ekstrem\/"},"modified":"2023-09-17T10:56:20","modified_gmt":"2023-09-17T10:56:20","slug":"maxima-minima-dari-suatu-fungsi-relatif-ekstrem","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/maxima-minima-dari-suatu-fungsi-relatif-ekstrem\/","title":{"rendered":"Maksimum dan minimum suatu fungsi (relatif ekstrem)"},"content":{"rendered":"<p>Pada artikel ini Anda akan menemukan cara menghitung maksimum dan minimum suatu fungsi, kami menjelaskannya kepada Anda dengan menyelesaikan dua contoh langkah demi langkah. Selain itu, Anda akan dapat berlatih dengan latihan langkah demi langkah tentang fungsi maksimum dan minimum. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-maximos-y-minimos-de-una-funcion\"><\/span> Berapakah maksimum dan minimum suatu fungsi?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Maksimum suatu fungsi adalah nilai terbesar dari fungsi tersebut dan minimum suatu fungsi adalah nilai terkecil dari fungsi tersebut.<\/strong> Maksima dan minima suatu fungsi disebut <strong>ekstrem relatif<\/strong> jika hanya mewakili nilai terbesar atau terkecil di lingkungannya, namun <strong>ekstrem absolut<\/strong> jika mewakili nilai terbesar atau terkecil dari keseluruhan fungsi. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/maximums-et-minimums-d-une-fonction.webp\" alt=\"maksimum dan minimum suatu fungsi\" class=\"wp-image-2437\" width=\"512\" height=\"420\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Anda juga dapat mengidentifikasi ekstrem relatif dengan mempelajari <strong>pertumbuhan dan penurunan fungsi<\/strong> :<\/p>\n<ul>\n<li> Suatu titik adalah <strong>maksimum relatif<\/strong> ketika fungsinya berubah dari naik ke turun.<\/li>\n<li> Suatu titik adalah <strong>nilai minimum relatif<\/strong> ketika suatu fungsi berubah dari menurun menjadi meningkat. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-los-maximos-y-minimos-de-una-funcion\"><\/span> Cara mencari nilai maksimum dan minimum suatu fungsi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dari turunan pertama dan kedua suatu fungsi, kita dapat mengetahui apakah suatu fungsi mempunyai ekstrem relatif pada suatu titik dan apakah titik tersebut merupakan maksimum relatif atau minimum relatif: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 10px; border-radius:30px;\">\n<ul style=\"color:#64B5F6; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Suatu fungsi mempunyai <strong>titik ekstrem terhadap<\/strong> titik-titik yang menghilangkan turunan pertamanya.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6dcc4b4bb7f26cf48a025c8e0dddf83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(a)=0 \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un extremo relativo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"357\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Dan tanda turunan kedua dari fungsi tersebut menentukan apakah titik tersebut maksimum atau minimum:<\/span>\n<ul style=\"color:#64B5F6; font-weight: bold; margin-left:7%; list-style-type:circle\">\n<li style=\"margin-bottom:10px\"> <span style=\"color:#101010;font-weight: normal;\">Jika turunan keduanya negatif, maka fungsi tersebut mempunyai <strong>maksimum relatif<\/strong> pada titik tersebut.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-211c91be3bfe6e5a91f048684198c70a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(a)<0 \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un m\\'aximo relativo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"360\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:10px\"> <span style=\"color:#101010;font-weight: normal;\">Jika turunan keduanya positif, maka fungsi tersebut memiliki <strong>minimum relatif<\/strong> pada titik tersebut.<\/span> <\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-089b7ae49fe440e3b4db19e0b17d8815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(a)>0 \\quad \\bm{\\longrightarrow} \\quad x=a \\text{ es un m\\&#8217;inimo relativo}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;356&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1-como-calcular-los-maximos-y-minimos-de-una-funcion\"><\/span> Contoh 1: Cara menghitung maksimum dan minimum suatu fungsi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Setelah kita melihat definisi maksimum dan minimum suatu fungsi, kita akan menyelesaikan contoh langkah demi langkah sehingga Anda dapat melihat cara menghitung maksimum dan minimum suatu fungsi.<\/p>\n<ul>\n<li> Hitunglah titik ekstrim relatif dari fungsi berikut dan tentukan apakah fungsi tersebut maksimum atau minimum:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d76dfe92202a4fa44057a7f4576c97a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"117\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Titik ekstrim relatif dari fungsi tersebut adalah titik-titik yang memenuhi<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Oleh karena itu, pertama-tama kita hitung turunan fungsi tersebut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cad79f4ba702585c8bece2546419bd83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x \\ \\longrightarrow \\ f'(x)=3x^2-3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"287\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dan sekarang kita menetapkan turunan dari fungsi tersebut sama dengan nol dan menyelesaikan persamaan kuadrat yang dihasilkan: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02ff903613a3329eb87c4943c4cb135b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2-3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"90\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-af1930d8be7faaf020a4103a17e484b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2=3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e8367eba5249bf4e7b1e395d86bb91be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=\\cfrac{3}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"52\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-959000af33497314f9a59a9bed2a19c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f0428d6e7e3932e00d3e6a7ab1a779d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, titik ekstrem relatif dari fungsi tersebut adalah x=+1 dan x=-1.<\/p>\n<p> Setelah kita mengetahui titik ekstrem relatif suatu fungsi, kita dapat mengetahui apakah fungsi tersebut maksimum atau minimum dengan tanda turunan keduanya. Oleh karena itu kami menghitung turunan kedua dari fungsi tersebut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2a906b229d6f4d4d03f59828f327fdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2-3 \\ \\longrightarrow \\ f''(x)=6x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"256\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dan sekarang kita mengevaluasi pada turunan kedua nilai ekstrim relatif yang kita temukan sebelumnya, untuk mengetahui apakah nilai tersebut merupakan maksimum atau minimum relatif:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb0c79376c0c8816492173e5f109809f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1)=6\\cdot 1 = 6 \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -5px;\"><\/p>\n<p> Minimal relatif<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8d51f54d437345293be122b03b5fff03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=6\\cdot (-1) = -6 \\ \\longrightarrow\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"222\" style=\"vertical-align: -5px;\"><\/p>\n<p> Relatif maksimal<\/p>\n<p> Turunan keduanya di x=1 adalah positif, jadi <strong>x=1 adalah minimum relatif<\/strong> . Sebaliknya, turunan kedua di x=-1 bernilai negatif, sehingga <strong>x=-1 merupakan maksimum relatif<\/strong> .<\/p>\n<p> Terakhir, kita substitusikan titik-titik yang ditemukan ke dalam fungsi asli untuk mencari koordinat Y dari titik ekstrem relatif:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65771f8b9ce10ad863604fe6e6dca867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=1^3-3\\cdot 1=-2 \\ \\longrightarrow \\ (1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"275\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-04df1e45775b1318795100d7211f3b32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=(-1)^3-3\\cdot(-1)= 2 \\ \\longrightarrow \\ (-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Kesimpulannya, fungsi ekstrem relatif adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimal untuk menunjuk<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80e956c137aedb103a56acc0cf510e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,-2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Maksimal tepat sasaran<\/strong> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92defeed12f15814813d53b8a24be9ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2-estudiar-la-monotonia-y-los-maximos-y-minimos-de-una-funcion\"><\/span> Contoh 2: Mempelajari monotonisitas serta maxima dan minima suatu fungsi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Sekarang mari kita lihat bagaimana jenis latihan lain diselesaikan. Dalam hal ini kami akan menjelaskan cara mencari maksimum dan minimum dari monotonisitas suatu fungsi.<\/p>\n<ul>\n<li> Pelajari monotonisitas dan hitung titik ekstrem relatif dari fungsi berikut:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb173dfd702785865be0051c9bcb7738_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2}{x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"101\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Hal pertama yang harus dilakukan adalah menghitung domain definisi fungsi. Sebagai fungsi rasional, kita perlu menetapkan penyebutnya sama dengan 0 untuk melihat bilangan mana yang tidak termasuk dalam domain fungsi tersebut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a57ca6c48b6f646aeb64eb7f05e4840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"73\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66d11e82f81cd2425ea2e6641e374baf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{1 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Setelah kita menghitung domain definisi fungsi, kita perlu mempelajari titik mana yang membatalkan turunan pertama. Oleh karena itu kami memperoleh fungsinya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-19cfa0164864d95d9b2d51743bf7c0d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2}{x-1} \\ \\longrightarrow \\ f'(x)= \\cfrac{2x\\cdot (x-1) - x^2\\cdot 1}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"364\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a280e333342dbe57e5d18839a1c9c0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{2x^2-2x - x^2}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"172\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c77f1f797549bb4663fca07fcea2302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{x^2-2x}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"128\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p> Dan sekarang kita menetapkan turunannya sama dengan 0 dan menyelesaikan persamaannya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-faf06fb85062e758f99800d1ffa0788b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2-2x}{\\left(x-1\\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"95\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p> Syarat<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc1f4cc53676f0eb98290b3478031fef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> Caranya adalah dengan membagi seluruh ruas kiri, sehingga kita dapat mengalikannya dengan seluruh ruas kanan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d8bb0359e60db0b26d9bfce1b349e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-2x=0\\cdot \\left(x-1\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-62138ee9fb8dc604ee836f1703379032_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-2x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kami mengekstrak faktor persekutuan untuk menyelesaikan persamaan kuadrat:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b243129a0d8853ec8716beb6d2d5c504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(x-2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Agar perkaliannya sama dengan 0, salah satu dari dua unsur perkaliannya harus nol. Oleh karena itu, kami menetapkan setiap faktor sama dengan 0 dan memperoleh dua solusi persamaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-55127e675ce8f7742db17d565c2ae507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(x-2) =0   \\longrightarrow  \\begin{cases} \\bm{x=0} \\\\[2ex] x-2=0 \\ \\longrightarrow \\ \\bm{x= 2} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"329\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Setelah kita menghitung domain dari fungsi dan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> , kami mewakili semua titik kritis yang ditemukan pada garis: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-0-1-2.webp\" alt=\"\" class=\"wp-image-2443\" width=\"399\" height=\"77\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Dan kita evaluasi tanda turunannya pada setiap interval, untuk mengetahui apakah fungsinya naik atau turun. Untuk melakukan hal ini, kita mengambil sebuah titik di setiap interval (tidak pernah titik kritisnya) dan melihat tanda apa yang dimiliki turunannya pada titik tersebut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c77f1f797549bb4663fca07fcea2302_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=\\cfrac{x^2-2x}{\\left(x-1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"128\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-171fa182722405650545d6e7fe14d5b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1) = \\cfrac{(-1)^2-2(-1)}{\\left((-1)-1\\right)^2} =\\cfrac{+3}{+4} = +0,75 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"369\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84e1013672adc10e9447af5478f592a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0,5) = \\cfrac{0,5^2-2\\cdot0,5}{\\left(0,5-1\\right)^2} = \\cfrac{-0,75}{+0,25} = -3  \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"363\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-110911aebecc81132e3d726e00be1fcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(1,5) = \\cfrac{1,5^2-2\\cdot1,5}{\\left(1,5-1\\right)^2} = \\cfrac{-0,75}{+0,25} = -3  \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"363\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0a995a54cd7d661f6431cdc3d0d0eda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3) = \\cfrac{3^2-2\\cdot3}{\\left(3-1\\right)^2} =\\cfrac{+3}{+4} = +0,75 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"313\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-0-1-2-positif-negatif-positif.webp\" alt=\"\" class=\"wp-image-2444\" width=\"400\" height=\"138\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Jika turunannya positif berarti fungsinya meningkat, tetapi jika turunannya negatif berarti fungsinya menurun. Oleh karena itu, interval pertumbuhan dan penurunannya adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Pertumbuhan:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11ebeca24ba262661dd73042a326110c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty, 0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Mengurangi:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-206ab3f38b17a58b25209bf269265919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,1)\\cup (1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Selanjutnya, pada x=0 fungsinya berubah dari naik ke turun, jadi <strong>x=0 adalah maksimum relatif<\/strong> dari fungsi tersebut <strong>.<\/strong> Dan pada x=2, fungsinya berubah dari menurun menjadi meningkat, jadi <strong>x=2 adalah minimum relatif<\/strong> dari fungsi tersebut.<\/p>\n<p> Dan terakhir, kita substitusikan titik-titik yang ditemukan pada fungsi asli untuk mencari koordinat Y dari ujung-ujungnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8bb02550f4c83abce02040f9e9ab495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=\\cfrac{0^2}{0-1} = \\cfrac{0}{-1} = 0 \\ \\longrightarrow \\ (0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"268\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-74333ede5561c728c68899d68b31ee62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2)=\\cfrac{2^2}{2-1} = \\cfrac{4}{1} = 4 \\ \\longrightarrow \\ (2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"254\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Singkatnya, fungsi ekstrem relatifnya adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Maksimal tepat sasaran<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c790019bd70403eba876c59c82c0f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimal untuk menunjuk<\/strong> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a59b564601b4cd9f2bc149baa80c44a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-maximos-y-minimos-de-una-funcion\"><\/span> Menyelesaikan latihan pada fungsi maksimum dan minimum<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung ekstrem relatif dari fungsi polinomial berikut dan tentukan apakah fungsi tersebut maksimum atau minimum: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-930d724a5ca23ed9152211f24dc2340b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x^2-9x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Titik ekstrem relatif suatu fungsi adalah titik di mana turunan pertama fungsi tersebut sama dengan nol. Oleh karena itu kami menghitung turunan dari fungsi tersebut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f353678aff2ff5f19c53042f35ef8a19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x^2-9x \\ \\longrightarrow \\  f'(x)=3x^2-6x-9\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"376\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita selesaikan persamaannya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58dcd049349f740f082d583dfd9e364c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77ac7ac1a36d5c8591235d8400eb68cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x^2-6x-9=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"131\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mempunyai persamaan kuadrat, jadi kami menerapkan rumus umum untuk menyelesaikannya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e4a1d5ede3779d54c8b9b66571a3394_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} x &amp;=\\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} =\\cfrac{-(-6) \\pm \\sqrt{(-6)^2-4\\cdot 3 \\cdot (-9)}}{2\\cdot 3}=\\\\[1.5ex]&amp;=\\cfrac{6 \\pm \\sqrt{144}}{6}=\\cfrac{6 \\pm 12}{6} =\\begin{cases} \\cfrac{6 + 12}{6}=\\cfrac{18}{6}= 3 \\\\[4ex] \\cfrac{6 - 12}{6}=\\cfrac{-6}{6}=-1 \\end{cases} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"168\" width=\"451\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, titik ekstrim relatif dari fungsi tersebut adalah titik x=3 dan x=-1.<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui titik ekstrem relatif suatu fungsi, kita dapat mengetahui apakah fungsi tersebut maksimum atau minimum dengan tanda turunan keduanya. Oleh karena itu kami membedakan fungsinya lagi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2ddaaae5740b93b84eb1db4c4e12f69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=3x^2-6x-9 \\ \\longrightarrow \\  f''(x)=6x-6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"327\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita mengevaluasi poin yang kita hitung sebelumnya pada turunan kedua: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b97f883eb74286ab41179d4353161816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)=6(3)-6=18-6 = +12 \\ \\longrightarrow \\ \\text{M\\'inimo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"365\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b6efbeb4ad4b54c03aa440dcafb7dc4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=6(-1)-6=-6-6 = -12 \\ \\longrightarrow \\ \\text{M\\'aximo}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"400\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Turunan keduanya di x=3 adalah positif, jadi <strong>x=3 adalah minimum<\/strong> . Dan turunan kedua di x=-1 bernilai negatif, jadi <strong>x=-1 maksimum<\/strong> .<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, kita substitusikan titik-titik yang ditemukan pada fungsi asli untuk mencari koordinat Y dari ujung-ujungnya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b885b81db85c9d12caeed0e046f14ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(3)=3^3-3\\cdot 3^2-9\\cdot3=-27 \\ \\longrightarrow \\ (3,-27)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"353\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-668346639ebe571949cd8e8939c8a4a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-1)=(-1)^3-3(-1)^2-9(-1)=5 \\ \\longrightarrow \\ (-1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"392\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Singkatnya, fungsi ekstrem relatifnya adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimum relatif terhadap intinya<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b572e4ffbdfe59c16e4e1a30b9ac82a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(3,-27)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Maksimum relatif terhadap intinya<\/strong> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c7aca1cad23e01f6998ce87ff4f73734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-1,5)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 2<\/h3>\n<p> Hitung ekstrem relatif dari fungsi eksponensial berikut dan tentukan apakah fungsi tersebut maksimum atau minimum: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e82a7f154d9620b6fdcd2d134cbf20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=e^x(x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, kita perlu membedakan fungsinya. Untuk melakukan ini, kami menerapkan rumus turunan suatu produk: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5508d7a8f5ef73fd09e2c8a013513229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=e^x\\cdot (x-1)+ e^x\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"206\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6d1f83cd5953e56070c9f8dea5a03ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=xe^x -e^x +e^x = xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita selesaikan persamaannya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58dcd049349f740f082d583dfd9e364c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9da9b4b19b1c7985bf785b693009de95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"xe^x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5c0d99b3aa4115c0415e0e57f5df2a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot e^x =0 \\longrightarrow \\begin{cases} \\bm{x=0} \\\\[2ex] e^x=0 \\ \\color{red}\\bm{\\times} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Suatu bilangan yang dipangkatkan ke bilangan lain tidak akan pernah menghasilkan 0. Oleh karena itu,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0108040ee23df4da2db681c9ffb2decc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"e^x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<p> tidak memiliki solusi dan satu-satunya solusi yang relatif ekstrim adalah<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita menghitung turunan kedua dari fungsi tersebut untuk mengetahui bahwa ekstrim relatif adalah maksimum atau minimum:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e3186824ec757b18335f7c6b93e6068_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= xe^x \\ \\longrightarrow \\ f''(x)= 1\\cdot e^x + x \\cdot e^x = e^x+xe^x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"394\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sekarang kita evaluasi pada turunan kedua ekstrim yang kita temukan sebelumnya, untuk melihat apakah maksimum atau minimum:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a333c6f36d372595070b5cf10ef06659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)= e^{0}+0\\cdot e^{0} = 1+0\\cdot 1 = 1 \\ \\longrightarrow \\ \\text{M\\'inimo}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"368\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena turunan keduanya di x=0 adalah positif, <strong>x=0 adalah minimum relatif atau lokal<\/strong> .<\/p>\n<p class=\"has-text-align-left\"> Terakhir, kita substitusikan titik yang ditemukan ke fungsi asli untuk mencari koordinat ujung lainnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b262d8c03601983b5497fc165bab677a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(0)=e^{0}(0-1) =1\\cdot (-1)=-1 \\ \\longrightarrow \\ (0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"357\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, satu-satunya ekstrem relatif dari fungsi tersebut adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimal untuk menunjuk<\/strong> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-737e35e6d1698a9e89986af90d34722e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 3<\/h3>\n<p> Pelajari monotonisitas dan temukan titik ekstrem relatif dari fungsi rasional berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03ea07dcfe35eeade4235b3325681c2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x -1 }{x^2+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"109\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, kita tentukan domain fungsinya. Untuk melakukan ini, kita menetapkan penyebut pecahan sama dengan nol dan menyelesaikan persamaan kuadrat yang dihasilkan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13670326c7cf3ae27c79e8e2ea4f438b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+1 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ekspresi<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b35b5141a240a76c5fc0e3c75ab5689d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -2px;\"><\/p>\n<p> Tidak akan pernah menjadi 0, karena hasil dari x <sup>2<\/sup> akan selalu berupa bilangan positif atau 0. Oleh karena itu, penjumlahan 1 tidak akan pernah menghasilkan 0. Oleh karena itu, domain dari fungsi tersebut hanya terdiri dari bilangan real:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f565027fd5d2a4381e3a23d183c9f76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Selanjutnya kita pelajari titik mana saja yang bertemu<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d9516fab46f301bc09e336a12418ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<p> Kami membedakan fungsi menggunakan aturan hasil bagi: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a21eceaf556455939314d569b69f365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x -1 }{x^2+1} \\ \\longrightarrow \\ f'(x)= \\cfrac{1 \\cdot (x^2+1) - (x-1) \\cdot 2x }{\\left(x^2+1}\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"415\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6104d51f83e54587e198db396734fec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{x^2+1-(2x^2-2x)}{\\left(x^2+1\\right)^2} = \\cfrac{x^2+1-2x^2+2x}{\\left(x^2+1\\right)^2}= \\cfrac{-x^2+2x+1}{\\left(x^2+1\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"515\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menetapkan turunannya sama dengan 0 dan menyelesaikan persamaan: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4890b9dfeb634c4d7a349351be73b5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-501383d3407e95ff1980351452e414f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-x^2+2x+1}{\\left(x^2+1\\right)^2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"143\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-348057b71ce15780c2f47bd8053e4cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-x^2+2x+1=0\\cdot \\left(x^2+1\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"219\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-579207dec3599e2925ad24d2e951cb47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-x^2+2x+1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mempunyai persamaan kuadrat, jadi kami menggunakan rumus umum untuk menyelesaikannya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-836d878f15098c1fe997fbb0392b8733_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}x &amp;=\\cfrac{-b \\pm \\sqrt{b^2-4ac}}{2a} =\\cfrac{-2 \\pm \\sqrt{2^2-4\\cdot (-1) \\cdot 1}}{2\\cdot (-1)} = \\\\[1.5ex]&amp;=\\cfrac{-2 \\pm \\sqrt{8}}{-2} =\\begin{cases} \\cfrac{-2 + \\sqrt{8}}{-2}= -0,41 \\\\[4ex] \\cfrac{-2 - \\sqrt{8}}{-2}= 2,41\\end{cases} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"174\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita menghitung domain dari fungsi dan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36700780d306ccf4975387990b1949fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> , kami mewakili semua titik tunggal yang ditemukan pada garis bilangan: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/number-line-041-241.webp\" alt=\"\" class=\"wp-image-2451\" width=\"319\" height=\"83\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Dan sekarang kita evaluasi tanda turunannya pada setiap interval, untuk mengetahui apakah fungsinya naik atau turun. Oleh karena itu, kita mengambil sebuah titik di setiap interval (bukan titik tunggalnya) dan melihat tanda apa yang dimiliki turunannya pada titik tersebut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c9e9690a2834e9ba455ebe711bfba4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)= \\cfrac{-(-1)^2+2(-1)+1}{\\left((-1)^2+1\\right)^2}}= \\cfrac{-2}{+4} =-0,5 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"412\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c280aa192cd61431df6a1ade0389ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(0)= \\cfrac{-0^2+2(0)+1}{\\left(0^2+1\\right)^2}}= \\cfrac{+1}{+1} =+1 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"340\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b40b424c7a763aa9849f33d850a10a1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(3)= \\cfrac{-3^2+2\\cdot 3+1}{\\left(3^2+1\\right)^2}}= \\cfrac{-2}{+100} =-0,02 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"375\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/ligne-numerique-041-241-negatif-positif-negatif.webp\" alt=\"\" class=\"wp-image-2453\" width=\"319\" height=\"150\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Jika turunannya positif berarti fungsi tersebut meningkat pada interval tersebut, tetapi jika turunannya negatif berarti fungsinya menurun. Oleh karena itu, interval pertumbuhan dan penurunannya adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Pertumbuhan:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-262e8d5f95ee4afe2dacc0037d8f334c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-0,41 \\ , \\ 2,41)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Mengurangi:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33c3a6bfd3dbfbdd2eff5fc4b70aea5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty \\ , \\ -0,41)\\cup (2,41 \\ , \\ +\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"231\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Fungsinya berubah dari turun ke naik pada x=-0,41, jadi <strong>x=-0,41 adalah minimum lokal<\/strong> dari fungsi tersebut. Dan fungsinya berubah dari naik ke turun pada x=2,41, jadi <strong>x=2,41 adalah maksimum lokal<\/strong> dari fungsi tersebut.<\/p>\n<p class=\"has-text-align-left\"> Terakhir, kita substitusikan nilai ekstrem yang ditemukan ke dalam fungsi asli untuk mencari koordinat Y dari titik-titik tersebut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-605a64bba8103c7ee0015a92b60273b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(-0,41)=\\cfrac{-0,41 -1 }{(-0,41)^2+1} = \\cfrac{-1,41}{1,17}= -1,21 \\ \\longrightarrow \\ (-0,41 \\ , \\ -1,21)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"532\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02ff38a48dc66cf3a658619cf41803c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(2,41)=\\cfrac{2,41 -1 }{2,41^2+1} = \\cfrac{1,41}{6,81}= 0,21 \\ \\longrightarrow \\ (2,41 \\ , \\ 0,21)\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"427\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, ekstrem relatif dari fungsi tersebut adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Minimal untuk menunjuk<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-906261d9a75f4bc2766c65fc0ac5a363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-0,41 \\ , \\ -1,21)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Maksimal tepat sasaran<\/strong> <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf7380f280bd665935068801c9c0d83d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{ (2,41 \\ , \\ 0,21)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 4<\/h3>\n<p> Kita tahu itu fungsinya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bc79cbc1c7886fe5d95d2db47d1635f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+ax+b\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<p> melewati titik tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f97f2fdc4d62902377daa83ebbd005b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -5px;\"><\/p>\n<p> dan mempunyai nilai yang relatif ekstrim<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d8c9d56ee018947d8f054cd237e8c06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1 .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\"><\/p>\n<p> Tentukan nilai yang tidak diketahui<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> dan nilai <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-73a3ea89ad967f2efadeb096bd87bdb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b .\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Biarkan fungsi tersebut memiliki ekstrem relatif<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee848c4f59793dbd8bd705b4e2411c8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> itu berarti sudah tercapai<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f57b701a1080acd4db5681249566b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(-1)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<p> Oleh karena itu, kami menghitung turunan dari fungsi tersebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee848c4f59793dbd8bd705b4e2411c8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= -1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<p> dan kami menetapkannya sama dengan 0: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41316f1389c40d8634eb0ad596956ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x^2+ax+b \\ \\longrightarrow \\ f'(x)=2x+a\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"309\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1786074f9a3b69a0c2a13a0db7a67895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f'(-1)=2(-1)+a\\\\[2ex] f'(-1)=0\\end{array} \\right\\} \\longrightarrow 2(-1)+a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kita selesaikan persamaan yang diperoleh untuk mencari nilai parameter a: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5e35d6b05179bb3e4db43f738b6da29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2(-1)+a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-729fe80252784c84b2a49624e59b2ac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2+a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"85\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-85d78ced37b5f76e83a3c9c24a8b3eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, fungsinya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbea1a44d5027753ebc196d004e5671d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+ax+b \\ \\xrightarrow{a \\ = \\ 2} \\ f(x)=x^2+2x+b\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"362\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Di sisi lain, mereka memberi tahu kita bahwa fungsi tersebut melalui suatu titik<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-702446e4df1457eff7e83e00a8709824_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,-2) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -5px;\"><\/p>\n<p> Artinya,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-304a45b518ffaec62b95f169ad647688_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(1)=-2 .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -5px;\"><\/p>\n<p> Oleh karena itu, kita dapat menerapkan kondisi ini untuk mencari nilai variabel b:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c09bb57a4a4fd3eb5d72f5d35d3c539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} f(1)=1^2+2\\cdot1+b \\\\[2ex] f(1)=-2 \\end{array} \\right\\} \\longrightarrow 1^2+2\\cdot 1+b = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"361\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kita selesaikan persamaan yang diperoleh untuk mencari nilai parameter b: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e05b252664d0ea2da72627e779469d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1^2+2\\cdot1+b=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"143\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-626f95581121d205b149c2323e711759_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1+2+b=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"113\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-625149278867f4929d813258055868e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=-2-1-2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"114\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e925fb7c5eaa30caa970c92688ede93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{b=-5}\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu fungsinya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2becf12662d9dc5f68cb13dd248f3e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2+2x+b \\ \\xrightarrow{b \\ = \\ -5} \\ f(x)=x^2+2x-5\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Pada artikel ini Anda akan menemukan cara menghitung maksimum dan minimum suatu fungsi, kami menjelaskannya kepada Anda dengan menyelesaikan dua contoh langkah demi langkah. Selain itu, Anda akan dapat berlatih dengan latihan langkah demi langkah tentang fungsi maksimum dan minimum. Berapakah maksimum dan minimum suatu fungsi? Maksimum suatu fungsi adalah nilai terbesar dari fungsi tersebut &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/maxima-minima-dari-suatu-fungsi-relatif-ekstrem\/\"> <span class=\"screen-reader-text\">Maksimum dan minimum suatu fungsi (relatif ekstrem)<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[49],"tags":[],"class_list":["post-43","post","type-post","status-publish","format-standard","hentry","category-representasi-fungsi"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fungsi maksimum dan minimum (relatif ekstrem) -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/id\/maxima-minima-dari-suatu-fungsi-relatif-ekstrem\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fungsi maksimum dan minimum (relatif ekstrem) -\" \/>\n<meta property=\"og:description\" content=\"Pada artikel ini Anda akan menemukan cara menghitung maksimum dan minimum suatu fungsi, kami menjelaskannya kepada Anda dengan menyelesaikan dua contoh langkah demi langkah. Selain itu, Anda akan dapat berlatih dengan latihan langkah demi langkah tentang fungsi maksimum dan minimum. Berapakah maksimum dan minimum suatu fungsi? 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Selain itu, Anda akan dapat berlatih dengan latihan langkah demi langkah tentang fungsi maksimum dan minimum. Berapakah maksimum dan minimum suatu fungsi? 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