Question

Find the maximum rate of change of f at the given point and the direction in which it occurs.

Answer 1

Step 1:

The function is and point is .

From gradient vector, .

Consider .

Apply partial derivative on each side with respect to .

Substitute in .

Apply partial derivative on each side with respect to .

Substitute in .

Then the gradient vector is .

Step 2:

The maximum rate of change of a function at a point occurs

in the direction is .

The magnitude of is the maximum rate of change.

The maximum rate of change of a function at a point occurs

in the direction is .

The magnitude of is .

Solution :

The maximum rate of change is 1.

The direction of the maximum rate of change is .