(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| cos θ . Since the minimum value of cos θ is -1 occurring, for 0 ≤ θ < 2π, when θ = , the minimum value of Du f is −|∇f|, occurring when the direction of u is the opposite of 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) = x4y − x2y3 decreases fastest at the point (2, −5).