Problem

Part I Multiple Choice

(A) If $f(x)$ is continuous on $(a,b)$, then there is $c\in(a,b)$ such that $f(c)=\max_{(a,b)}f(x)$.

(B) Suppose that $f(x)$ is continuous but not differentiable at $x=0$. The function $g(x)=xf(x)$ must be differentiable at $x=0$.

(C) If $f(x)$ is a differentiable function, then $\lim_{x\to a}f'(x)=f'(a)$.

(D) If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is a unique $c\in(a,b)$ such that $f(b)-f(a)=f'(c)(b-a)$.