Classify each critical point as a relative minimum, relative maximum or saddle points.

Question

Find all critical points for following functions. Classify each critical point as a relative minimum, relative maximum or saddle points.

1.f(x, y) = 4xy − 10x^2 − 4y^2 + 8x + 8y − 10

2. f(x, y) = x^3 − 27x − y^3 + 3y + 7

Answer 1

(1)

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 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 maximum at .

Relative maximum at .

Solution:

Relative maximum at .

Answer 2

(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 points are and  .

Comments

Contd...

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 .