Find all points where the function has any relative extrema or saddle points and identify the type of relative extreme.

Question

y/x +2/y + x?

Answer 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.

Answer 2

contd...

Step 4 :

Find the critical points :

Equate to zero.

Equate to zero.

Substitute in the above equation

When , .

The critical point are and .

Step 5 :

Find the quantity of d .

The quantity of d  at .

Find at  .

Since  and , then f  has a relative maximum at .

The quantity of d  at .

Find  at .

If and , then f  has a relative minimum at .

Solution :

Relative maximum at .

Relative minimum at .