Steepest Ascent and Steepest Descent

Definition (Steepest Ascent-Descent) The direction of the greatest rate of increase (or decrease) of a given function at a specified point is called the direction of steepest ascent (or steepest descent).

Proposition (Steepest Ascent-Descent) Suppose is differentiable at the point and that the gradient of at satisfies Then

    (i)   The largest value of the directional derivative at is and occurs when the unit vector points in the direction of

    (ii)   The smallest value of the directional derivative at is and occurs when the unit vector points in the direction of

    Proof. If is any unit vector, then

where is the angle between and But assumes its largest value of 1 at that is, when points in the direction Thus, the largest possible value of is Also assumes its smallest value when This value occurs when points toward and in this direction

Example (Steepest Ascent-Descent) Find the steepest ascent or steepest descent.

(a) In what direction is the function defined by increasing most rapidly at the point , and what is the maximum rate of increase? In what direction is decreasing most rapidly?

    Solution. We begin by finding the gradient of

At ,

The most rapid rate of increase is and occurs in the direction of   The most rapid rate of decrease occurs in the direction of and is

(b) Let At the point find the unit vector pointing in the direction of most rapid increase of

    Solution. The gradient is

Thus, and so

(c) Find the maximum rate of change of at the point and the direction in which it occurs.

    Solution. We compute

Thus the maximum rate of change is and occurs in the direction of

(d) Find the maximum rate of change of at the point and the direction in which it occurs.

    Solution. We compute

Thus the maximum rate of change is

steepest ascent and steepest descent _gr_68.gif]

and occurs in the direction of