Interesting Problem:

Show that $f(x,y) = x^2 +4y^2 - 4xy +2$ has an infinite number of critical points and that $D = 0 $ at each one. Then show that $f$ has a local (and absolute) minimum at each critical point.

Visual/estimation

Use level curves to estimate the local mins, maxes, and saddle points of the function. Then use calculus to find these values precisely.

• $f(x,y) = x^2 + y^2 + x^{-2}y{-2}$
• $f(x,y) = xy e^{-x^2-y^2}$

Absolute mins and max on a region

Find the absolute max and mins of $f$ on the set $D$

• $f(x,y) = 3 + xy - x -2y$, $D$ is the closed triangular region wtih verticies $(1,0), (5,0)$, and $(1,4)$.
• $f(x,y) = x^3-3x-y^3 + 12 y$, $D$ is the quadrilateral whose vertices are $(-2,3),(2,3),(2,2)$ and $(-2,-2)$.