Chapter 13.9, Problem 2ES

The accompanying figure shows the graphs of the line $3x+4y=25$ and the level curves of height $c=9,16,25,36,\text{and}49$ for the function $f\left(x,y\right)=x^2+y^2.$

(a) Use the accompanying figure to find the minimum value of the function $f\left(x,y\right)=x^2+y^2$ subject to $3x+4y=25,$ and explain your reasoning.

(b) How can you tell from the accompanying figure that your answer to part (a) is not the maximum value of f subject to the constraint?

(c) Use Lagrange multipliers to check your work.