Classify the critical point $(x,y)=(\frac{}{6},\frac{}{5})$ of the function

$f(x,y)=cos(6x+5y)$

The range of $f(x,y)$ is $-1,1$ and $f(\frac{}{6},\frac{}{5})=1,$ so the critical point is the location of a maximum.

The range of $f(x,y)$ is $-1,1$ and $f(\frac{}{6},\frac{}{5})=-1,$ so the critical point is the location of a minimum.

The range of $f(x,y)$ is $-1,1$ and $f(\frac{}{6},\frac{}{5})=0,$ so the critical point is the location of a saddle point.

$f_{}(\frac{}{6},\frac{}{5})f_{yy}(\frac{}{6},\frac{}{5})-f_{xy}^2(\frac{}{6},\frac{}{5})>0$ and

$f_{}(\frac{}{6},\frac{}{5})<mn>0</mn>$ so$,$ by the second derivative test, the critical point is the location of a maximum.

$f_{}(\frac{}{6},\frac{}{5})f_{yy}(\frac{}{6},\frac{}{5})-f_{xy}^2(\frac{}{6},\frac{}{5})>0$ and

$f_{}(\frac{}{6},\frac{}{5})>0$ so$,$ by the second derivative test, the critical point is the location of a minimum.

the

second derivative test, the critical point is the location of a saddle point.