Question

(a) 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 θ be the angle between ∇f(x) and unit vector u. Then Du f = |∇f| Correct: Your answer is correct. . Since the minimum value of Correct: Your answer is correct. is -1 Correct: Your answer is correct. occurring, for 0 ≤ θ < 2π, when θ = Correct: Your answer is correct. , the minimum value of Du f is −|∇f|, occurring when the direction of u is Correct: Your answer is correct. the direction of ∇f (assuming ∇f is not zero).

(b) Use the result of part (a) to find the direction in which the function f(x, y) = x3y − x2y3 decreases fastest at the point (3, −4).

Answer 1

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