Step 1 :

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 .  

Step 2 : 

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.

Step 3 : 

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.

Step 4 :

Find the critical points :

Equate   to zero.

Equate to zero.

Substitute in equation (1).

Substitute in equation (1).

The critical points are and .

Step 5 :

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 .

Step 6 :

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

Solution :

The local minimum is and