Let V be open in Rn, a V, f V R, and suppose that f is differentiable at a a) Prove that the directional derivative Duf(a) exists (see Exercise 11 2 10) for each u Rn such that u 1 and Duf(a) f(a) u b) If f(a) 0 and represents the angle between u and f(a), prove that Duf(a) f(a) cos c) Show that as u ranges over all unit vectors in Rn, the maximum of Duf(a) is f(a) , and it occurs when u is parallel to f(a)

Question: Let V be open in Rn, a V, f : V R, and suppose that f is differentiable at a. a) Prove that

Let V be open in Rn, a ∈ V, f : V → R, and suppose that f is differentiable at a.
a) Prove that the directional derivative Duf(a) exists (see Exercise 11.2.10) for each u ∈ Rn such that ||u|| = 1 and Duf(a) = f(a) ∙ u.
b) If f(a) ≠ 0 and θ represents the angle between u and f(a), prove that Duf(a) = ||f(a)|| cos θ.
c) Show that as u ranges over all unit vectors in Rn, the maximum of Duf(a) is ||f(a)||, and it occurs when u is parallel to f(a).

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