College Board AP Classroom Unit 5 Progress Check: MCQ Part B 10 11 12 Question 5 A Let gbe the function defined by g (z) = (z? – 1 + 1)e What in the absolute maximum value of g on the interval -4,1) O Type here to search<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The problem is asking for the absolute maximum value of the function ( g(z) = (z^2 – 1 + 1)e^z ) on the interval ( [-4, 1] ).<\/p>\n\n\n\n The given function is:<\/p>\n\n\n\n [ Simplifying the expression inside the parentheses:<\/p>\n\n\n\n [ To find the absolute maximum, we first need to take the derivative of the function and find the critical points within the interval ( [-4, 1] ).<\/p>\n\n\n\n The derivative of ( g(z) = z^2 e^z ) is found using the product rule:<\/p>\n\n\n\n [ To find the critical points, set ( g'(z) = 0 ):<\/p>\n\n\n\n [ Since ( e^z ) is never zero, we focus on solving:<\/p>\n\n\n\n [ This gives two critical points: ( z = 0 ) and ( z = -2 ).<\/p>\n\n\n\n Now we evaluate ( g(z) ) at the critical points and the endpoints of the interval ( [-4, 1] ).<\/p>\n\n\n\n Now we compare the values:<\/p>\n\n\n\n The absolute maximum value of ( g(z) ) on the interval ( [-4, 1] ) is ( g(1) = e ).<\/p>\n\n\n\n The absolute maximum value of ( g(z) ) on the interval ( [-4, 1] ) is ( e \\approx 2.7183 ).<\/p>\n","protected":false},"excerpt":{"rendered":" College Board AP Classroom Unit 5 Progress Check: MCQ Part B 10 11 12 Question 5 A Let gbe the function defined by g (z) = (z? – 1 + 1)e What in the absolute maximum value of g on the interval -4,1) O Type here to search The Correct Answer and Explanation is : […]<\/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-181357","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181357","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=181357"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181357\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=181357"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=181357"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=181357"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Simplify the function<\/h3>\n\n\n\n
g(z) = (z^2 – 1 + 1)e^z
]<\/p>\n\n\n\n
g(z) = z^2 e^z
]<\/p>\n\n\n\nStep 2: Find the critical points<\/h3>\n\n\n\n
g'(z) = \\frac{d}{dz}(z^2) \\cdot e^z + z^2 \\cdot \\frac{d}{dz}(e^z)
]
[
g'(z) = 2z e^z + z^2 e^z
]
Factor out ( e^z ):
[
g'(z) = e^z(2z + z^2)
]<\/p>\n\n\n\n
e^z(2z + z^2) = 0
]<\/p>\n\n\n\n
2z + z^2 = 0
]
[
z(2 + z) = 0
]<\/p>\n\n\n\nStep 3: Evaluate the function at the critical points and endpoints<\/h3>\n\n\n\n
\n
[
g(-4) = (-4)^2 e^{-4} = 16 e^{-4}
]<\/li>\n\n\n\n
[
g(-2) = (-2)^2 e^{-2} = 4 e^{-2}
]<\/li>\n\n\n\n
[
g(0) = 0^2 e^0 = 0
]<\/li>\n\n\n\n
[
g(1) = 1^2 e^1 = e
]<\/li>\n<\/ol>\n\n\n\nStep 4: Determine the maximum value<\/h3>\n\n\n\n
\n
Final Answer:<\/h3>\n\n\n\n