5. Let r > 0,

a, bE R, f : Br

(a,

b) ---t R be differentiable, and (x, y) E Br

(a, b).

(a) Compute the derivative of g(t) = f(tx + (1 - t)a, y) + f

(a, ty + (1 - t)b).

(b) Prove that there are numbers c between a and x, and d between band y, such that f(x, y) - f

(a,

b) = (x - a)fx

(c, y) + (y - b)fy

(a, d).

(This is Exercise 12.20 in Apostol [1].)