{"id":44,"date":"2023-09-17T10:56:06","date_gmt":"2023-09-17T10:56:06","guid":{"rendered":"https:\/\/mathority.org\/id\/fungsi-kecekungan-dan-kecembungan-suatu-kelengkungan\/"},"modified":"2023-09-17T10:56:06","modified_gmt":"2023-09-17T10:56:06","slug":"fungsi-kecekungan-dan-kecembungan-suatu-kelengkungan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/fungsi-kecekungan-dan-kecembungan-suatu-kelengkungan\/","title":{"rendered":"Kecekungan dan konveksitas suatu fungsi (kelengkungan)"},"content":{"rendered":"<p>Di sini Anda akan mempelajari apa itu kecekungan dan kecembungan suatu fungsi dan cara mengetahui apakah suatu fungsi cekung atau cembung. Selain itu, Anda akan dapat berlatih dengan latihan langkah demi langkah tentang kelengkungan suatu fungsi. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-concavidad-y-la-convexidad-de-una-funcion\"><\/span> Berapakah kecekungan dan kecembungan suatu fungsi?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Kecekungan dan konveksitas suatu fungsi mengacu pada kelengkungan grafik suatu fungsi.<\/strong> <strong>Fungsi cekung<\/strong> adalah fungsi yang grafiknya berbentuk gunung, dan <strong>fungsi cembung<\/strong> adalah fungsi yang grafiknya berbentuk lembah.<\/p>\n<p> Pada paragraf sebelumnya, fungsi cekung dan fungsi cembung telah didefinisikan secara informal untuk memudahkan pemahaman, namun definisi matematis dari fungsi cekung dan fungsi cembung adalah sebagai berikut:<\/p>\n<ul>\n<li> <strong>Fungsi cekung:<\/strong> bila ruas yang menghubungkan dua titik mana pun dari fungsi tersebut berada di bawah kurva.<\/li>\n<li> <strong>Fungsi cembung:<\/strong> bila ruas yang menghubungkan dua titik mana pun dari fungsi tersebut berada di atas kurva. <\/li>\n<\/ul>\n<div class=\"wp-block-columns is-layout-flex wp-container-9\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\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\/fonction-concave.webp\" alt=\"fungsi cekung\" class=\"wp-image-2556\" width=\"292\" height=\"300\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\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\/fonction-convexe.webp\" alt=\"fungsi cembung\" class=\"wp-image-2557\" width=\"288\" height=\"300\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<p> Pada akhirnya, perbedaan antara fungsi cekung dan fungsi cembung terletak pada bentuk fungsinya sehingga, Anda dapat membedakan kecekungan dan kecembungan dari grafik fungsinya.<\/p>\n<p> Namun, suatu fungsi tidak harus cekung atau cembung pada seluruh domainnya, tetapi bisa juga cekung pada satu interval dan cembung pada interval lainnya.<\/p>\n<p class=\"has-background\" style=\"background-color:#fffde7\"> <strong>Catatan:<\/strong> Komunitas matematika masih belum sepenuhnya setuju dan, oleh karena itu, beberapa profesor mengatakan sebaliknya: mereka menyebut suatu fungsi cekung yang berbentuk 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-f5ebc563dbe58138d1de6b7fe99e8d31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\cup}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> , dan fungsi cembung yang berbentuk<strong> <\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dccbfcebef91876585ebd365457c3d24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\cap}\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p>. Pokoknya yang penting tahu apa fungsinya, apapun namanya. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-estudiar-la-curvatura-de-una-funcion\"><\/span> Cara mempelajari kelengkungan suatu fungsi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Mempelajari kelengkungan suatu fungsi melibatkan pencarian kecekungan dan kecembungan suatu fungsi, yaitu mengetahui interval di mana fungsi tersebut cekung dan interval di mana fungsi tersebut cembung.<\/p>\n<p> Jadi, untuk mempelajari kelengkungan suatu fungsi, harus dilakukan langkah-langkah sebagai berikut: <\/p>\n<div style=\"background-color:#FFF3E0; padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 10px; border-radius:30px;\">\n<ol style=\"color:#64B5F6; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Temukan <strong>titik-titik yang tidak termasuk dalam domain<\/strong> fungsi tersebut.<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Hitung turunan pertama dan <strong>turunan kedua fungsi tersebut.<\/strong><\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Temukan <strong>akar-akar turunan kedua<\/strong> , yaitu menghitung titik-titik yang menghilangkan turunan kedua dengan menyelesaikannya\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> .<\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Buatlah <strong>interval<\/strong> dengan akar-akar turunan dan titik-titik yang tidak termasuk dalam domain fungsi tersebut.<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Hitung nilai turunan kedua pada suatu titik pada setiap interval.<\/span><\/li>\n<li style=\"margin-bottom:16px\"> <span style=\"color:#101010;font-weight: normal;\">Jadi, <strong>tanda turunan kedua<\/strong> menentukan kecekungan atau kecembungan suatu fungsi pada interval ini:<\/span>\n<ul style=\"color:#64B5F6; font-weight: bold; margin-top:8px; margin-left:8%\">\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Jika turunan kedua dari fungsi tersebut positif, maka fungsi tersebut <strong>cembung<\/strong> pada interval tersebut.<\/span><\/li>\n<li style=\"margin-bottom:8px\"> <span style=\"color:#101010;font-weight: normal;\">Jika turunan kedua dari fungsi tersebut negatif, maka fungsi tersebut <strong>cekung<\/strong> pada interval tersebut.<\/span> <\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-hallar-la-curvatura-de-una-funcion\"><\/span> Contoh cara mencari kelengkungan suatu fungsi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Selanjutnya, kita akan menyelesaikan contoh langkah demi langkah sehingga Anda dapat melihat bagaimana interval kecekungan dan kecembungan suatu fungsi dihitung.<\/p>\n<ul>\n<li> Pelajari kecekungan dan kecembungan 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-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> Hal pertama yang harus dilakukan adalah menghitung domain definisi fungsi. Dalam hal ini, kita mempunyai fungsi polinomial, sehingga domain dari fungsi tersebut 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> Setelah kita menghitung domain dari fungsi tersebut, kita perlu menyelidiki pada titik mana turunan kedua dari fungsi tersebut hilang.<\/p>\n<p> Oleh karena itu kami menghitung turunan pertama 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-5e356b083f137690f09e2af3c62b8b07_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> Selanjutnya kita mencari 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-016a210c39feb9d6df8caddacdbac681_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 menetapkan turunan kedua 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-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-7878c5d5f08c25729d37b94ea643bdf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-108341259f39e597d1e8c926a80dbae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{0}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"45\" 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-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>\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-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -5px;\"><\/p>\n<p> , kami mewakili semua titik kritis yang ditemukan di garis. Dalam hal ini kami tidak menemukan titik kritis dalam perhitungan domain definisi fungsi, tetapi kami memperoleh titik yang membatalkan turunan kedua fungsi tersebut: <\/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\/nombre-ligne-0.webp\" alt=\"\" class=\"wp-image-2426\" width=\"201\" height=\"77\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Dan sekarang kita evaluasi tanda turunan keduanya pada setiap interval, untuk mengetahui apakah fungsinya cekung atau cembung. Oleh karena itu, kita mengambil sebuah titik pada setiap interval (tidak pernah titik kritisnya) dan melihat tanda apa yang dimiliki turunan kedua 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-86d5e98dcde33659284eaafb55050852_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=6x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-ab7d8a0d32ae8b29c211f51359c61a4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1) = 6\\cdot (-1)=-6 \\  \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"236\" 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-084869779374d781df1af76f0aa784ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1) = 6\\cdot 1=+6 \\  \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/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\/droite-numerique-0-concave-convexe.webp\" alt=\"\" class=\"wp-image-2561\" width=\"201\" height=\"134\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Terakhir, kami menyimpulkan interval kecekungan dan kecembungan fungsi tersebut. Jika turunan keduanya positif berarti fungsinya cembung.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , dan jika turunan keduanya negatif berarti fungsinya cekung<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Jadi interval kecekungan dan kecembungan fungsi tersebut adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Cembung<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91ffae1f3397d9d66e3159e131554abb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(0,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Cekung<\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong> <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bec51dbadb80b73308bcb5a625cd152f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-concavidad-y-convexidad-de-una-funcion\"><\/span> Latihan soal kecekungan dan kecembungan suatu fungsi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung interval kecekungan dan kecembungan fungsi polinomial 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-9cd1ae6bdb4d8dbd44ef2561f6e022fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) = x^3-3x^2-2x\" 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\"> Fungsi dalam latihan ini adalah polinomial, sehingga domain dari fungsi tersebut 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\"> Setelah menentukan domain suatu fungsi, kita bedakan:<\/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-0a14436352ca08e03441a24b9e881093_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^3-3x^2-2x \\ \\longrightarrow \\  f'(x)= 3x^2-6x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"375\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Selanjutnya kita mencari 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-8ec8542448a8f4aa326a2d70f5f5efb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= 3x^2-6x-2 \\ \\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 menetapkan turunan kedua 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-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-5c644c35f109a535e50e1bda440f12db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x-6= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" 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-7b7a2a9a8deaab9c4eb4761838f99407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x= 6\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-8ddd3a5381ed7b33d4e9dfbc270c1da3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\cfrac{6}{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"45\" 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-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-left\"> Setelah kita menghitung domain dari fungsi tersebut dan 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-981d85a257dd56afdb3fc7eb53d5eadf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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\/nombre-ligne-1.webp\" alt=\"\" class=\"wp-image-2564\" width=\"213\" height=\"83\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Dan sekarang mari kita ambil suatu titik yang termasuk dalam setiap interval dan lihat tanda apa yang memiliki turunan kedua pada titik ini:<\/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-4e4f2f534e4134661dfda2b6b8125ee3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(0)= 6\\cdot 0-6 = -6 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" 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-59d344f0c0b93695edf55652112bf6e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(2)= 6\\cdot 2-6 = 12-6=+6 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"297\" style=\"vertical-align: -5px;\"><\/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\/droite-numerique-1-concave-convexe.webp\" alt=\"\" class=\"wp-image-2565\" width=\"224\" height=\"150\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Jika turunan keduanya lebih besar dari nol berarti fungsinya cembung.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , tetapi bila turunan keduanya negatif berarti fungsinya cekung<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Oleh karena itu, interval kecekungan dan kecembungan adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Cembung<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87d6843b66a0ebea6c769017a30a8d75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(1,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Cekung<\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong> <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b85798b30f125fea3702a0671c77ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" 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> Pelajari kelengkungan 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-3ac9ccc5e8540cca38f599ed36507792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x^3}{x^2-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"109\" style=\"vertical-align: -12px;\"><\/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 menghitung domain dari fungsi tersebut. Karena ini adalah fungsi rasional, kita menetapkan penyebutnya sama dengan nol 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-a34dfe78d673534873a2013c16e1b353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-4= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"81\" 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-c269e23a1070b3e5556abece040af75a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2=4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" 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-c4c32359b264b28ac80f2606c09d5a2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^2}=\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" 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-c06a55e3acdd1e283973786926b27716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\pm 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Artinya jika x adalah -2 atau +2, penyebutnya adalah 0. Oleh karena itu, fungsinya tidak akan ada. Oleh karena itu, domain fungsi tersebut terdiri dari semua bilangan kecuali x=-2 dan x=+2.<\/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-e666f828709575f965b5120fbdda085e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f= \\mathbb{R}-\\{-2, +2 \\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kedua, kita menghitung turunan pertama 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-1397d0e8e73bd7b1d851411dee28daed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^3}{x^2-4}  \\ \\longrightarrow \\ f'(x)= \\cfrac{3x^2 \\cdot (x^2-4) - x^3 \\cdot 2x }{\\left(x^2-4\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"396\" 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-495f08a881718b2734ef1db17b5f39ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)= \\cfrac{3x^4-12x^2-2x^4}{\\left(x^2-4\\right)^2} = \\cfrac{x^4-12x^2}{\\left(x^2-4\\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"298\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Lalu kita selesaikan turunan keduanya: <\/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-50df15bb48cacf8f031b640994661e47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^2 - \\left(x^4-12x^2\\right)\\cdot 2\\left(x^2-4\\right)\\cdot 2x }{ \\left(\\left(x^2-4\\right)^2 \\right)^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"483\" style=\"vertical-align: -33px;\"><\/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-3b05b5f09c2adbfead593df2cdf2ad29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^2 - \\left(x^4-12x^2\\right)\\cdot 4x\\left(x^2-4\\right) }{\\left(x^2-4\\right)^4 }\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"461\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Semua suku dikalikan dengan<\/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-1e0f0d9a63183e28c50a5cedcddeddd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x^2-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"60\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Oleh karena itu kita dapat menyederhanakan pecahan 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-92e1aa280d06bf8b58045845d5e21f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right)^{\\cancel{2}} - \\left(x^4-12x^2\\right)\\cdot 4x\\cancel{\\left(x^2-4\\right)} }{\\left(x^2-4\\right)^{\\cancelto{3}{4}} }\" title=\"Rendered by QuickLaTeX.com\" height=\"52\" width=\"458\" 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-a912eff3359969b6ffbef96a3f16932d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{\\left(4x^3-24x\\right)\\cdot \\left(x^2-4\\right) - \\left(x^4-12x^2\\right)\\cdot 4x}{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"386\" 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-8bc35bdd2b70bbac52fa0f24bbefa261_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{4x^5-16x^3-24x^3+96x - \\left(4x^5-48x^3\\right) }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"381\" 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-045971f71cc11ced77ea0df9f2c514fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{4x^5-16x^3-24x^3+96x - 4x^5+48x^3 }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"365\" 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-3144a0aa00ee8ec427752f05f0fac40c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= \\cfrac{8x^3+96x  }{\\left(x^2-4\\right)^3 }\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"145\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung akar-akar 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-f618f4961c18c45be60fc496ad4896e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(x)= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" 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-e8ed519add27a4d51c75b49179e632ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{8x^3+96x  }{\\left(x^2-4\\right)^3 }=0\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"109\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> 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-d8590a90fab5aef55d7b45cd89e01943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(x^2-4\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"70\" style=\"vertical-align: -7px;\"><\/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-b5c0d1e44accc3b68a67598f5c4d834c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8x^3+96x =0\\cdot \\left(x^2-4\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"193\" 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-4c52a7448e4acc67488ef5747cc3bed9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"8x^3+96x =0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mengekstrak faktor persekutuan:<\/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-4f2884a1c9b8dabf6ea5323f2ac71b2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x(8x^2+96)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"123\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agar perkaliannya sama dengan 0, salah satu dari dua unsur perkaliannya harus nol. Oleh karena itu, kami menetapkan setiap faktor 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-31adba554b44aa92fd7227506440ccaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x\\cdot(8x^2+96) =0 \\longrightarrow \\begin{cases} \\bm{x =0} \\\\[2ex] 8x^2+96=0 \\ \\longrightarrow \\ x^2=\\cfrac{-96}{8}} = -12 \\ \\longrightarrow \\ x= \\sqrt{-12} \\ \\color{red}\\bm{\\times} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"635\" 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-635e2ffa452a5a66a4bcacb0e111c5ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x= \\sqrt{-12}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -3px;\"><\/p>\n<p> Tidak ada penyelesaian karena tidak ada akar negatif dari bilangan real.<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita nyatakan pada garis semua titik kritis yang diperoleh, yaitu titik-titik yang tidak termasuk dalam domain (x=-2 dan x=+2) dan titik-titik yang membatalkan turunan kedua (x=0): <\/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\/droite-numerique-2-0-2.webp\" alt=\"\" class=\"wp-image-2399\" width=\"385\" height=\"75\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Dan kita evaluasi tanda turunan keduanya pada setiap interval, untuk mengetahui apakah fungsinya cekung atau cembung. Jadi kita ambil sebuah titik di setiap interval dan lihat tanda apa yang mempunyai turunan kedua di 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-3b5618d1ab96a078d50507f45155595b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-3)=\\cfrac{8(-3)^3+96(-3)  }{\\left((-3)^2-4\\right)^3 } = \\cfrac{-504}{125}=-4,03 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"408\" 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-e2e868f1f815d4155a187c55b004cc13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(-1)=\\cfrac{8(-1)^3+96(-1)  }{\\left((-1)^2-4\\right)^3 } = \\cfrac{-104}{-27}=3,85 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"394\" 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-3814746f7f9e8aa3920e3f84cd0ff0eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(1)=\\cfrac{8\\cdot1^3+96\\cdot 1  }{\\left(1^2-4\\right)^3 } = \\cfrac{104}{-27}=-3,85 \\ \\rightarrow \\ \\bm{-}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"348\" 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-54d98824f72954de12bc065471a610e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)=\\cfrac{8\\cdot 3^3+96\\cdot 3  }{\\left(3^2-4\\right)^3 } = \\cfrac{504}{125}=4,03 \\ \\rightarrow \\ \\bm{+}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"329\" 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-2-0-2-courbure.webp\" alt=\"\" class=\"wp-image-2568\" width=\"383\" height=\"129\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p class=\"has-text-align-left\"> Jika turunan keduanya positif berarti fungsinya cembung.<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> , dan jika turunan keduanya negatif berarti fungsinya cekung<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> . Oleh karena itu, interval kecekungan dan kecembungan adalah:<\/p>\n<p class=\"has-text-align-center\"> <strong>Cembung<\/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-49efa6d9ab88562f20df743cb7d267f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cup})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong><\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cd739761b2dc845594c0a0696a240c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-2,0)\\cup (2,+\\infty)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> <strong>Cekung<\/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-59e636042d77445b1534260d9d7309a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\bm{\\cap})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> <strong>:<\/strong> <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e741ac026627200772655094f921f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(-\\infty,-2)\\cup (0,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" 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> Sebuah fungsi<\/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-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> memiliki relatif ekstrim dalam<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Terlebih lagi, fungsinya cembung<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5b03f70f64c85542b93152492ab8bd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\cup )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> pada titik yang sama ini. Tentukan apakah ekstrim relatifnya adalah minimum atau maksimum.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/maxima-minima-dari-suatu-fungsi-relatif-ekstrem\/\">definisi maxima dan minima suatu fungsi<\/a><\/span> <\/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 cembung<\/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-d2bb599fff4b55075f6de7628a35f822_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\cup)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"><\/p>\n<p> Di dalam<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> berarti turunan kedua pada titik ini positif, yaitu<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e1c4b76e92a6fd3dd8c16dfb2ca8cf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)>0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;19&#8243; width=&#8221;74&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<p> .<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, relatif ekstrim dari<\/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-2176aa2efb23d1fbd644eff672465ff0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> <strong>Ini adalah minimum<\/strong> , karena<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8cfc4aa0fa7cede2c69561e8b65a6991_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f''(3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -5px;\"><\/p>\n<p> Itu positif.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Di sini Anda akan mempelajari apa itu kecekungan dan kecembungan suatu fungsi dan cara mengetahui apakah suatu fungsi cekung atau cembung. Selain itu, Anda akan dapat berlatih dengan latihan langkah demi langkah tentang kelengkungan suatu fungsi. Berapakah kecekungan dan kecembungan suatu fungsi? Kecekungan dan konveksitas suatu fungsi mengacu pada kelengkungan grafik suatu fungsi. Fungsi cekung &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/fungsi-kecekungan-dan-kecembungan-suatu-kelengkungan\/\"> <span class=\"screen-reader-text\">Kecekungan dan konveksitas suatu fungsi (kelengkungan)<\/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":[38],"tags":[],"class_list":["post-44","post","type-post","status-publish","format-standard","hentry","category-derivatif"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Cekungan dan konveksitas suatu fungsi (kelengkungan)<\/title>\n<meta name=\"description\" content=\"Kami menjelaskan cara mempelajari kecekungan dan kecembungan (kelengkungan) suatu fungsi. 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