Exercise 34

Respond to each of the following prompts to solve the given optimization problems.

1. Let $f(x,y) = \sin(x)+\cos(y)$. Determine the absolute maximum and minimum values of $f$. At what points do these extreme values occur?

2. For a certain differentiable function $F$ of two variables $x$ and $y$, its partial derivatives are $$\begin{gathered} F_x(x,y) = x^2-y-4 \ \ \mbox{and} \ \ F_y(x,y) = -x + y - 2. \end{gathered}$$ Find each of the critical points of $F$, and classify each as a local maximum, local minimum, or a saddle point.

3. Determine all critical points of $T(x,y) = 48 + 3xy - x^2y - xy^2$ and classify each as a local maximum, local minimum, or saddle point.

4. Find and classify all critical points of $g(x,y) = \frac{x^2}{2} + 3y^3 + 9y^2 - 3xy + 9y - 9x$

5. Find and classify all critical points of $z = f(x,y) = ye^{-x^2-2y^2}$.

6. Determine the absolute maximum and absolute minimum of $f(x,y) = 2 + 2x + 2y - x^2 - y^2$ on the triangular plate in the first quadrant bounded by the lines $x = 0$, $y = 0$, and $y = 9-x$.

7. Determine the absolute maximum and absolute minimum of $f(x,y) = 2 + 2x + 2y - x^2 - y^2$ over the closed disk of points $(x,y)$ such that $(x-1)^2 + (y-1)^2 \le 1$.

8. Find the point on the plane $z = 6 - 3x - 2y$ that lies closest to the origin.