Maximum And Minimum Values

(requires JavaScript)

1. Suppose that $\left(0,2\right)$ is a critical point of a function $g$ with continuous second partial derivatives. In each case, what can you say about $g$?

   1. $g_{xx}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=-1,g_{xy}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=6,g_{yy}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=1$
   2. $g_{xx}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=-1,g_{xy}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=2,g_{yy}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=-8$
   3. $g_{xx}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=4,g_{xy}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=6,g_{yy}<!--FUNCTION\; APPLICATION-->\left(0,2\right)=9$

2. Find local extrema and saddle points of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=9-2x+4y-x^2-4y^2$

   Maximum at $\left(-1,\frac{1}{2},11\right)$.

3. Find local extrema and saddle points of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x^4+y^4-4xy+2$

   Minima at $\left(1,1,0\right)$ and $\left(-1,-1,0\right)$, saddle point at $\left(0,0,2\right)$.

4. Find local extrema and saddle points of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\left(2x-x^2\right)\left(2y-y^2\right)$

   There are $5$ critical points. There is a local maximum at $\left(1,1\right)$ and saddle nodes at $\left(0,0\right)$, $\left(0,2\right)$, $\left(2,0\right)$, and $\left(2,2\right)$.

5. Find the absolute maximum and minimum values of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=1+4x-5y$

   on the closed triangular region with vertices $\left(0,0\right)$, $\left(2,0\right)$, and $\left(0,3\right)$.

   Maximum at $\left(2,0,9\right)$ and minimum at $\left(0,3,-14\right)$.

6. Find the absolute maximum and minimum values of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=3+xy-x-2y$

   on the closed triangular region with vertices $\left(1,0\right)$, $\left(5,0\right)$, and $\left(1,4\right)$.

7. Find the absolute maximum and minimum values of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x^2+y^2+x^{2}y+4$

   on the region $\left\{\left(x,y\right)|\left|x\right|\le 1,\left|y\right|\le 1\right\}$.

   Maxima at $\left(1,1,7\right)$ and $\left(-1,1,7\right)$, minimum at $\left(0,0,4\right)$.
8. Find the points on the surface $y^2=9+xz$ that are closest to the origin.
9. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $x+2y+3z=6$.
   ${\displaystyle \frac{4}{3}}$