(811, #23) Show that a differentiable function decreases most rapidly at x in the direction opposite to the gradient vector, that is, in the direction of .

Solution: From the proof of Theorem 15 we have:

where is the angle between and u, and u is a unit vector. The maximum value occurs when , .

The minimum value occurs when , .

Thus the function decreases most rapidly in the direction opposite to the gradient vector.

No Maple Code for this problem.

by: gm