Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration

Q. 27

Found in: Page 985


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.


Short Answer

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

f(x,y)=xy when x2+4y2=16

The maximum value of the function is 4 and the minimum value is -4 and both exist as the constraint is a bounded and closed ellipse.

See the step by step solution

Step by Step Solution

Step 1. Given information.   

Given function is f(x,y)=xy.

Given constraint is x2+4y2=16.

Step 2. critical points of the function. 

Gradients of function.


Use the method of Lagrange multipliers.


Compare terms.

y=2λxλ=y2xx=8λyλ=x8yso x2=4y2

substitute x2=4y2 in constraint.

role="math" localid="1649893915369" x2+4y2=164y2+4y2=168y2=16y=±2x=±22

so critical points are role="math" localid="1649893960368" -22,-2,22,-2 ,-22,2, & 22,2.

Step 3. maximum and minimum of a function. 

Find function value at -22,-2,22,-2 ,-22,2, & 22,2.




So the maximum value of the function is 4 and the minimum value is -4.

As constraint is bounded and closed ellipse so maximum and minimum both exist.

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.