Mathematics, 30.03.2021 20:00 Guest4747

(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(theta) . Since the minimum value of cos(theta) is -1 occurring, for 0 ≤ θ < 2π, when θ = π , 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) = x^(3)y − x^(2)y^(3) decreases fastest at the point (4, −4)

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(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the...