Exercises

01

Classify critical points on topographical contour map

02

Analyze two critical points

Let $f(x,y)=2x^2-8xy+y^4-4y^3$. Find the two critical points of $f$ and analyze them using the discriminant test. You are encouraged to check your results by graphing the function in Demos3d or Math3d. What is the minimum value taken by $f$?

03

Maximum on closed domain

Find the global maximum value of $y^2+xy-x^2$ on the square $x\in [0,2]$, $y\in [0,2]$. Where does this occur? Remember to check critical points on the interior and optimize the function on the boundary.

04

Biggest box

Find the largest volume of a box with nonnegative coordinates, all but one vertex on the coordinate planes, and the remaining vertex lying in the paraboloid given by $\frac{x^2}{4}+\frac{y^2}{9}+z=1$. (So one corner is at the origin, and the opposite corner is on the paraboloid.) What are the dimensions of this box?