2. At which points on the curve y = 1 + 40x3<\/sup> – 3x5<\/sup> does the tangent line have the largest slope?<\/p>\n\n\n\n (x,y) Smaller x-value<\/p>\n\n\n\n (x,y) Larger x-value<\/p>\n\n\n\n The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n Let’s solve each problem step by step.<\/p>\n\n\n\n Given the function: y=1+40×3\u22123x5y = 1 + 40x^3 – 3x^5<\/p>\n\n\n\n The slope of the tangent line is given by the derivative: dydx=120×2\u221215×4\\frac{dy}{dx} = 120x^2 – 15x^4<\/p>\n\n\n\n To find the points where the tangent has the largest slope<\/strong>, we differentiate again to find critical points of dy\/dxdy\/dx: d2ydx2=240x\u221260×3\\frac{d^2y}{dx^2} = 240x – 60x^3<\/p>\n\n\n\n Set the second derivative equal to zero: 240x\u221260×3=0240x – 60x^3 = 0<\/p>\n\n\n\n Factor: 60x(4\u2212x2)=060x(4 – x^2) = 0<\/p>\n\n\n\n Solve for xx: x=0orx=\u00b12x = 0 \\quad \\text{or} \\quad x = \\pm2<\/p>\n\n\n\n To determine where the slope is maximum, we check d2y\/dx2d^2y\/dx^2 at these points. The largest positive value of d2y\/dx2d^2y\/dx^2 occurs at x=\u22122x = -2 and x=2x = 2, indicating maximum slopes.<\/p>\n\n\n\n Find the corresponding y-values<\/strong>: y(\u22122)=1+40(\u22122)3\u22123(\u22122)5=1+40(\u22128)\u22123(\u221232)y(-2) = 1 + 40(-2)^3 – 3(-2)^5 = 1 + 40(-8) – 3(-32) =1\u2212320+96=\u2212223= 1 – 320 + 96 = -223 y(2)=1+40(2)3\u22123(2)5=1+40(8)\u22123(32)y(2) = 1 + 40(2)^3 – 3(2)^5 = 1 + 40(8) – 3(32) =1+320\u221296=225= 1 + 320 – 96 = 225<\/p>\n\n\n\n Thus, the points where the tangent has the largest slope are:<\/p>\n\n\n\n Now, I will generate an image of the function y=1+40×3\u22123x5y = 1 + 40x^3 – 3x^5 showing the points where the maximum slope occurs.<\/p>\n\n\n\n Here is the graph of the function y=1+40×3\u22123x5y = 1 + 40x^3 – 3x^5, highlighting the points (\u22122,\u2212223)(-2, -223) and (2,225)(2, 225) where the tangent line has the largest slope. Let me know if you need any further explanation!<\/p>\n\n\n\n 1. A walkway is 4 feet wide and is paved with square stones that are eight inches wide. How many stones wide is the path? 2. At which points on the curve y = 1 + 40×3 – 3×5 does the tangent line have the largest slope? (x,y) Smaller x-value (x,y) Larger x-value The correct answer and […]<\/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-196219","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196219","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=196219"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196219\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Problem 1: Walkway Width in Stones<\/strong><\/h3>\n\n\n\n
\n
\n\n\n\nProblem 2: Finding Points with Maximum Slope<\/strong><\/h3>\n\n\n\n
\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-129-1024x791.png\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"