Find max/min values.

You want to equate the first partials to zero and solve the resulting system to find the critical points:

$f_x(x,y)=0$

$f_y(x,y)=0$

Solve the resulting system to find any critical points in the given region.

Now you want to use the second partials test for relative extrema:

Let $(a,b)$ be a critical point of $z=f(x,y)$ and suppose $f_{xx},\,f_{yy}$ and $f_{xy}$ are continuous in a rectangular region containing $(a,b)$ .

Let $D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-(f_{xy}(x,y))^2$ .

i) If $D(a,b)>0$ and $f_{xx}(a,b)>0$ , then $f(a,b)$ is a relative minimum.

ii) If $D(a,b)>0$ and $f_{xx}(a,b)<0$ , then $f(a,b)$ is a relative maximum.

iii) If $D(a,b)<0$ , then $f(a,b)$ is not an extremum.

iv) If $D(a,b)=0$ , then no conclusion can be drawn concerning a relative extremum.