(1)

Step 1 :

Second partials 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 .

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 the above equation.

Substitute in .

The critical point is .

Step 5 :

Find the quantity of d .

Find at  .

If and , then f  has a relative minimum at .

Relative maximum at .

Solution :

Relative maximum at .

(2)

Step 1 :

Second partial 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 then is a saddle point.

4. The test is inconclusive if .

Step 2 :

 The function 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.

The critical point are and  .

Step 5 :

Find the quantity of d .

Case(i): Consider critical point as .

As , then f  has a saddle point at .

Case(ii): Consider critical point as .

Find at  .

If and , then f  has a relative maximum at .

Relative maximum at .

Case(iii): Consider critical point as .

Find at  .

If and , then f  has a relative minimum at .

Relative minimum at .

Case(iv): Consider critical point as .

As , then f  has a saddle point at .

Solution :

Relative maximum at .

Relative minimum at .

Saddle points are and .