Let r > 0, a, b R, f: Br(a, b) R be differentiable, and (x,

Question:

Let r > 0, a, b ∈ R, f: Br(a, b) → R be differentiable, and (x, y) ∈ Br(a, b).
a) Let g(t) = f(tx + (1 - t)a, y) + f{a, ty + (1 - t)b) and compute the derivative of g.
b) Prove that there are numbers c between a and x, and d between b and y such that
f(x, y) - f(a, b) = {x - a)fx(c, y) + (y - b)fy(a, d).