Final Exam Review
.pdf
School
Pennsylvania State University**We aren't endorsed by this school
Course
MATH 231
Subject
Mathematics
Date
Dec 19, 2024
Pages
1
Uploaded by AgentWaterCheetah17
Math 231Multivariable Calculus and Vector AnalysisSection 15Final Exam Review: April 22The following problems are practice on the concepts we’ve seen in Chapter 14. Problemsmarked with ( ) are more difficult than the average problem.DO NOT PANIC IF YOUCAN’T DO A PROBLEM WITH A ( ).14.71.Find and classify all points of localandabsolute maxima and minima off(x, y) =e−(x2+y2+2x)in the diskD:={(x, y) : (x+ 1)2+y2≤1}.2.Find and classify all points of localandabsolute maxima and minima off(x, y) =x2+y2−2y+ 1 in the diskD:={(x, y) :x2+y2≤4}.3.Fixnpoints (a1, b1), . . . ,(an, bn).Show that the sum of the squares of the distancesfromP= (c, d) to each of thenfixed points is minimized whencis the average of theaianddis the average of thebi.4.( ) We prove Young’s Inequality.Namely, for allx, y≥0 andα≥1 andβ≥1with1α+1β= 1, we have thatxy≤xαα+xββWe do so by provingf(x, y) =xαα+xββ−xyis non-negative for allx, y≥0.a) Show that the set of critical points off(x, y) is the curvey=xα−1. Note that this curvecan also be described asx=yβ−1. What is the value off(x, y) at points on this curve?b) Verify that the Second Derivative Test (the multivariate version) fails. Show, however,that for fixedb >0, the functiong(x) =f(x, b) is concave up with a critical point atx=bβ−1.c) Conclude that for allx >0,f(x, b)≥f(bβ−1, b)−0.14.85. Find the maximum and minimum values of the functionf(x, y) = 4x2+ 9y2subject tothe constraintxy= 4.6. Find the maximum and minimum values of the functionf(x, y) =x2y4subject to theconstraintx2+ 2y2= 1.7.Find the maximum and minimum values of the functionf(x, y) =xy+ 2zsubjectto the constraintx2+y2+z2= 36.8.Find the maximum and minimum values of the functionf(x, y) =xy+xzsubjectto the constraintx2+y2+z2= 4.Pennsylvania State University1
