I've been having a bit of trouble with the last part of the following question:
Let $a\in[A,B], f\in C^2[A,B]$, and let $P_1(x) = f(a) + f^{\prime}(a)(x-a)$ be the first-order Taylor polynomial. Fix a point $x_0\in[A,B].$
(a) Define $h(t) = f(t) + f^{\prime}(t)(x_0-t) + A(x_0-t)^2$, find the constant $A$ that makes $h(x_0) = h(a)$.
(b) Apply Rolle's Theorem to $h$ to obtain a point $c$ between $x_0$ and $a$ such that $$f(x_0) - P_1(x_0) = f^{\prime\prime}(c)\frac{(x_0-a)^2}{2}$$
(c) Find a constant $M$ so that $|f(x)-f(a)|\leq M(x-a)^2$
The first two parts are pretty straightforward, but I don't see how to solve (c). You can use (b) to bound the difference $f(x) - P_1(x)$ as desired, but I don't see a clear way to modify this to get just $f(x)-f(a)$ bounded. I thought there might be a way to use the Generalized MVT for this, but I don't see a clear way for that either. Any help would be appreciated!