I am taking a Calculus II course in college, and because we learned about power series today, I've been trying to work out a proof for Taylor's Remainder Theorem as a practice exercise. However, I've been stuck on part (b) for a while now, and I'm getting quite frustrated.
My solution for part (a):
$$f(x)-f(a) = \int_a^x f'(t) dx$$ $$f(x) = f(a) + \int_a^x f'(t) dx$$
My attempt for part (b) using integration by parts:
$$f(x) = f(a) + f'(x)x-f'(a)a-\int_a^x f''(t)t dt$$
I'm not sure where to go from there, or if my integration is even correct in that step. Can someone please at least give me a hint?