Question

Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of −∇f(x). Let theta be the angle between ∇f(x) and unit vector u. Then Du f = |∇f| . Since the minimum value of :. occurring, for 0 ≤ theta < 2, when theta = π , the minimum value of Du f is −|∇f|, occurring when the direction of u is the direction of ∇f (assuming ∇f is not zero).

Answer 1

A differentiable function decreases most rapidly in the direction opposite the gradient vector -∇f(x), with the minimum value of Du f occurring when the direction of u aligns with ∇f.

Step-by-step explanation:

To show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, we can consider the directional derivative Du f where u is a unit vector. The angle θ between the gradient vector ∇f(x) and u can be used to find the value of Du f. If we let θ = π (opposite direction), then the minimum value of Du f is -|∇f| when the direction of u is also opposite to ∇f(x).