7. Let V be open in Rn, a E V, f: V -+ R, and let f be differentiable at a.

(a) Prove that the directional derivative Duf

(a) exists (see Exercise 9, p. 338), for each u E Rn such that Ilull = 1, and Duf

(a) = V' f

(a) . u.

(b) If V'f

(a) -I- 0 and e represents the angle between u and V'f

(a) , prove that Duf

(a) = IIV'f(a)llcose.

(c) Show that as u ranges over all unit vectors in Rn, the maximum of Duf(a)

is IIV' f

(a) II, and it occurs when u is parallel to V' f(a).