Question

Examine the function for relative extrema and saddle points.

g (x, y) = xy

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 , 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.

The critical point is .

Find the quantity d :

Since , is a saddle point.

Substitute the point in .

The saddle point is .

Solution :

The saddle point is .