{"id":167319,"date":"2024-11-15T15:45:51","date_gmt":"2024-11-15T15:45:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=167319"},"modified":"2024-11-15T15:45:53","modified_gmt":"2024-11-15T15:45:53","slug":"find-all-the-points-where-the-fumction-is-not-differentiable-fx%e2%88%a5x%e2%88%a5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/15\/find-all-the-points-where-the-fumction-is-not-differentiable-fx%e2%88%a5x%e2%88%a5\/","title":{"rendered":"Find all the points where the fumction is not differentiable: f(x)=\u2225x\u2225"},"content":{"rendered":"\n<p>Find all the points where the fumction is not differentiable: f(x)=\u2225x\u2225<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The function ( f(x) = |x| ) refers to the absolute value of ( x ), where ( |x| = |x| ). This is a piecewise function defined as:<\/p>\n\n\n\n<p>[<br>f(x) =<br>\\begin{cases}<br>x, &amp; \\text{if } x \\geq 0, \\<br>-x, &amp; \\text{if } x &lt; 0.<br>\\end{cases}<br>]<\/p>\n\n\n\n<p>To find the points where the function is not differentiable, we need to consider where the function might have sharp corners or discontinuities, as these are typical places where differentiability fails.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Check the function at ( x = 0 )<\/h3>\n\n\n\n<p>The function ( f(x) = |x| ) has a sharp corner at ( x = 0 ). At this point, the left-hand and right-hand derivatives are not equal:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Right-hand derivative<\/strong>: For ( x > 0 ), the function ( f(x) = x ). The derivative of ( f(x) ) is ( f'(x) = 1 ) for ( x > 0 ).<\/li>\n\n\n\n<li><strong>Left-hand derivative<\/strong>: For ( x &lt; 0 ), the function ( f(x) = -x ). The derivative of ( f(x) ) is ( f'(x) = -1 ) for ( x &lt; 0 ).<\/li>\n<\/ul>\n\n\n\n<p>At ( x = 0 ), the right-hand derivative is ( 1 ) and the left-hand derivative is ( -1 ). Since the left-hand and right-hand derivatives do not match, the function is <strong>not differentiable at ( x = 0 )<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Check the function for other points<\/h3>\n\n\n\n<p>For any ( x \\neq 0 ), the function ( f(x) = |x| ) behaves as either ( f(x) = x ) or ( f(x) = -x ), both of which are differentiable. Specifically:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For ( x > 0 ), ( f(x) = x ) is differentiable with derivative ( f'(x) = 1 ).<\/li>\n\n\n\n<li>For ( x &lt; 0 ), ( f(x) = -x ) is differentiable with derivative ( f'(x) = -1 ).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>The function ( f(x) = |x| ) is differentiable for all ( x \\neq 0 ), but it is <strong>not differentiable at ( x = 0 )<\/strong> due to the discontinuity in the derivative at that point. Therefore, the function is not differentiable only at ( x = 0 ).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find all the points where the fumction is not differentiable: f(x)=\u2225x\u2225 The Correct Answer and Explanation is: The function ( f(x) = |x| ) refers to the absolute value of ( x ), where ( |x| = |x| ). This is a piecewise function defined as: [f(x) =\\begin{cases}x, &amp; \\text{if } x \\geq 0, \\-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-167319","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/167319","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=167319"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/167319\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=167319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=167319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=167319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}