Second derivatives test :

If f  have continuous partial derivatives on an open region containing a point  for which   

 and .

To test for relative extrema of f , consider the quantity

1. If  and , then f  has a relative minimum at .

2. If  and , then f  has a relative maximum at .

3. If  and then  is a saddle point. 

4. The test is inconclusive if .

The function is .

The domain is 

Apply partial derivative on each side with respect to x.

Differentiate  partially with respect to x.

Differentiate  partially with respect to y.

The function is 

Apply partial derivative on each side with respect to y

Differentiate  partially with respect to y.

Differentiate  partially with respect to x.

Find the critical points :

Equate   to zero.

Equate  to zero.

Substitute  in equation (1).

Substitute  in equation (1).

The critical points are  and .

Find the value of f  at the critical points :

Find the quantity D : 

At the point .

Since  and , the function f  has a local minimum at .

Substitute the point  in .

The local minimum is 

At the point .

Since , the graph has saddle point at .

Find the value of f  at the boundary points :

The domain of the function is .

Find the quantity D : 

At the point .

Since , the graph has saddle point at .

At the point .

Since  and , the function f  has a local minimum at . 

Substitute the point  in .

The local minimum is 

The local minimum is  and