Let r > 0, (a, b) R2, f: Br(a, b) R, and suppose that the first-order partial derivatives fx and fy exist in Br(a, b)

Let r > 0, (a, b) ˆˆ R2, f: Br(a, b) †’ R, and suppose that the first-order partial derivatives fx and fy exist in Br(a, b) and are differentiable at (a, b).
a) Set Δ(h) = f(a + h, b + h) - f(a + h, b) - f(a, b + h) + f(a, b) and prove for h sufficiently small that

for some t ˆˆ (0, 1).
b) Prove that

c) Prove that

A(h) lab+ th)-ya.b)-V,la.b) (h, th) (f,a, bth) f(a, b) -Vfy(a, b 0, th+hfyx (a, b) (11) 2 Fyx (a, b). lim as-av (a , b) = a vax (a, b). dx dy