Which of the following is a true statement?<\/p>\n\n\n\n
Question 1 ?Answer<\/p>\n\n\n\n
a.<\/p>\n\n\n\n
The function h(x)=-x^2<\/p>\n\n\n\n
has an absolute minimum at x=0<\/p>\n\n\n\n
.<\/p>\n\n\n\n
b.<\/p>\n\n\n\n
The function l(x)=|x+2|<\/p>\n\n\n\n
is differentiable at x=-2<\/p>\n\n\n\n
. ?<\/p>\n\n\n\n
c.<\/p>\n\n\n\n
The function q(x)=x^4<\/p>\n\n\n\n
is decreasing for all values of x<\/p>\n\n\n\n
.<\/p>\n\n\n\n
d.<\/p>\n\n\n\n
The function f(x)=-x^3<\/p>\n\n\n\n
has no critical points.<\/p>\n\n\n\n
e.<\/p>\n\n\n\n
The function p(x)=sin(x)<\/p>\n\n\n\n
has infinitely many critical points.<\/p>\n\n\n\n
f.<\/p>\n\n\n\n
The function g(x)=-cos(x)<\/p>\n\n\n\n
has no critical points.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The correct statement is:<\/p>\n\n\n\n e. The function p(x) = sin(x) has infinitely many critical points.<\/strong><\/p>\n\n\n\n Let’s analyze each option in detail:<\/p>\n\n\n\n a. The function h(x) = -x\u00b2 has an absolute minimum at x = 0.<\/strong><\/p>\n\n\n\n This is false<\/strong>. The function ( h(x) = -x^2 ) is a downward-opening parabola, which means it has an absolute maximum at ( x = 0 ), not a minimum. As ( x ) moves away from zero, the function values become increasingly negative, so no minimum exists at ( x = 0 ).<\/p>\n\n\n\n b. The function l(x) = |x + 2| is differentiable at x = -2.<\/strong><\/p>\n\n\n\n This is false<\/strong>. The function ( l(x) = |x + 2| ) has a “corner” at ( x = -2 ), which means it is not differentiable there. The derivative from the left does not equal the derivative from the right at this point, leading to a discontinuity in the slope.<\/p>\n\n\n\n c. The function q(x) = x\u2074 is decreasing for all values of x.<\/strong><\/p>\n\n\n\n This is false<\/strong>. The function ( q(x) = x^4 ) is increasing for ( x > 0 ) and decreasing for ( x < 0 ). At ( x = 0 ), the function is flat, but it is not globally decreasing. The behavior depends on the sign of ( x ).<\/p>\n\n\n\n d. The function f(x) = -x\u00b3 has no critical points.<\/strong><\/p>\n\n\n\n This is false<\/strong>. The function ( f(x) = -x\u00b3 ) does have a critical point at ( x = 0 ). A critical point occurs when the derivative is zero or undefined. The derivative of ( f(x) = -x\u00b3 ) is ( f'(x) = -3x\u00b2 ), which equals zero at ( x = 0 ), so this is a critical point.<\/p>\n\n\n\n e. The function p(x) = sin(x) has infinitely many critical points.<\/strong><\/p>\n\n\n\n This is true<\/strong>. The function ( p(x) = \\sin(x) ) has critical points where the derivative is zero. The derivative of ( \\sin(x) ) is ( \\cos(x) ), and ( \\cos(x) = 0 ) at ( x = \\frac{\\pi}{2} + n\\pi ), where ( n ) is any integer. Hence, there are infinitely many critical points of the form ( x = \\frac{\\pi}{2} + n\\pi ).<\/p>\n\n\n\n f. The function g(x) = -cos(x) has no critical points.<\/strong><\/p>\n\n\n\n This is false<\/strong>. The derivative of ( g(x) = -\\cos(x) ) is ( g'(x) = \\sin(x) ). ( g'(x) = 0 ) at ( x = n\\pi ), where ( n ) is an integer, so there are critical points at these locations.<\/p>\n\n\n\n The correct answer is e.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":" Which of the following is a true statement? Question 1 ?Answer a. The function h(x)=-x^2 has an absolute minimum at x=0 . b. The function l(x)=|x+2| is differentiable at x=-2 . ? c. The function q(x)=x^4 is decreasing for all values of x . d. The function f(x)=-x^3 has no critical points. e. The function […]<\/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-188657","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188657","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=188657"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188657\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188657"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188657"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Explanation:<\/h3>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n