{"id":163413,"date":"2024-11-08T19:13:51","date_gmt":"2024-11-08T19:13:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=163413"},"modified":"2024-11-08T19:13:53","modified_gmt":"2024-11-08T19:13:53","slug":"can-a-function-be-differentiable-at-a-horizontal-tangent","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/08\/can-a-function-be-differentiable-at-a-horizontal-tangent\/","title":{"rendered":"Can a function be differentiable at a horizontal tangent"},"content":{"rendered":"\n<p>Can a function be differentiable at a horizontal tangent?<\/p>\n\n\n\n<p>a) Yes<br>b) No<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>The answer is:<\/p>\n\n\n\n<p><strong>a) Yes<\/strong><\/p>\n\n\n\n<p>A function can indeed be differentiable at a horizontal tangent. To understand why, let\u2019s break down what differentiability and a horizontal tangent mean.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Differentiability<\/strong>: A function is differentiable at a point if it has a defined derivative at that point, meaning the function has a well-defined rate of change at that point. Differentiability implies that the function is smooth and has no sharp corners, jumps, or vertical tangents at the point in question.<\/li>\n\n\n\n<li><strong>Horizontal Tangent<\/strong>: A horizontal tangent line at a point on the function\u2019s graph means that the slope of the tangent line at that point is zero. In calculus, if ( f(x) ) has a horizontal tangent at a point ( x = c ), then the derivative ( f'(c) = 0 ).<\/li>\n<\/ol>\n\n\n\n<p>A classic example is the function ( f(x) = x^2 ). The derivative of ( f(x) = x^2 ) is ( f'(x) = 2x ). At ( x = 0 ), the derivative ( f'(0) = 0 ), which indicates a horizontal tangent line at that point. However, the function is differentiable at ( x = 0 ) because the derivative exists and is continuous there.<\/p>\n\n\n\n<p>So, the presence of a horizontal tangent simply indicates that the slope (derivative) is zero at that point, not that the derivative is undefined. Differentiability depends on the function being smooth and having a well-defined derivative, not on the value of the derivative itself. A function can have points where the derivative is zero (horizontal tangents) while still being differentiable at those points.<\/p>\n\n\n\n<p>Therefore, the correct answer is <strong>Yes, a function can be differentiable at a point where it has a horizontal tangent.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Can a function be differentiable at a horizontal tangent? a) Yesb) No The Correct Answer and Explanation is : The answer is: a) Yes A function can indeed be differentiable at a horizontal tangent. To understand why, let\u2019s break down what differentiability and a horizontal tangent mean. A classic example is the function ( f(x) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","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":"","ast-disable-related-posts":"","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-163413","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/163413","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=163413"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/163413\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=163413"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=163413"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=163413"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}