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

0 votes
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
asked Oct 18, 2015 in CALCULUS by anonymous

2 Answers

0 votes

(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 image.

Apply partial derivative on each side with respect to x.

image

Differentiate partially with respect to x.

image

Differentiate partially with respect to y.

image

Step 3:

The function is image.

Apply partial derivative on each side with respect to y.

image

Differentiate partially with respect to y.

image

Differentiate partially with respect to x.

image

Step 4:

Find the critical points.

Equate to zero.

image

Equate to zero.

image

Substitute image in the above equation.

image

Substitute image in image.

image

The critical point is image.

Step 5:

Find the quantity of d.

image

Find image at  image.

image

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

Relative maximum at image.

Solution:

Relative maximum at image.

answered Oct 19, 2015 by Lucy Mentor
edited Oct 19, 2015 by steve
0 votes

(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.

image

Differentiate partially with respect to y.

image

Differentiate partially with respect to x.

image

Step 4:

Find the critical points.

Equate to zero.

image

Equate to zero.

image

The critical points are image and  image.

answered Oct 19, 2015 by cameron Mentor

Contd...

Step 5:

Find the quantity of d .

image

Case(i): Consider critical point as image.

image

As , then f  has a saddle point at image.

Case(ii): Consider critical point as image.

image

Find image at  image.

image

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

Relative maximum at image.

Case(iii): Consider critical point as image.

image

Find image at  image.

image

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

Relative minimum at image.

Case(iv): Consider critical point as image.

image

As , then f  has a saddle point at image.

Solution:

Relative maximum at image.

Relative minimum at image.

Saddle points are image and image.

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