(a) State the Intermediate Value Theorem.

(b) Define what it means for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ to be differentiable at a point $a \in \mathbb{R}$. If $f$ is differentiable everywhere on $\mathbb{R}$, must $f^{\prime}$ be continuous everywhere? Justify your answer.

State the Mean Value Theorem.

(c) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable everywhere. Let $a, b \in \mathbb{R}$ with $a<b$.

If $f^{\prime}(a) \leqslant y \leqslant f^{\prime}(b)$, prove that there exists $c \in[a, b]$ such that $f^{\prime}(c)=y$. [Hint: consider the function $g$ defined by

$$g(x)=\frac{f(x)-f(a)}{x-a}$$

if $x \neq a$ and $\left.g(a)=f^{\prime}(a) .\right]$

If additionally $f(a) \leqslant 0 \leqslant f(b)$, deduce that there exists $d \in[a, b]$ such that $f^{\prime}(d)+f(d)=y$.