Chapter 13, Problem 5PS

Finding Maximum and Minimum Values

(a) Let $f(x,y)=x-y$ and $g(x,y)=x^2+y^2=4$. Graph various level curves of $f$ ad the constraint $g$ in the $xy$ -plane. Use the graph to determine the maximum value of $f$ subject to the constraint $g=4$. Then verify your answer using Lagrange multipliers.

(b) Let $f(x,y)=x-y$ and $g(x,y)=x^2+y^2=0$. Find the maximum and minimum values of $f$ subject to the constraint $g=0$. Does the Method of Lagrange Multipliers work in this case? Explain.