State and prove the Intermediate Value Theorem.

Suppose that the function $f$ is differentiable everywhere in some open interval containing $[a, b]$, and that $f^{\prime}(a)<k<f^{\prime}(b)$. By considering the functions $g$ and $h$ defined by

$$g(x)=\frac{f(x)-f(a)}{x-a} \quad(a<x \leqslant b), \quad g(a)=f^{\prime}(a)$$

and

$$h(x)=\frac{f(b)-f(x)}{b-x} \quad(a \leqslant x<b), \quad h(b)=f^{\prime}(b),$$

or otherwise, show that there is a subinterval $\left[a^{\prime}, b^{\prime}\right] \subseteq[a, b]$ such that

$$\frac{f\left(b^{\prime}\right)-f\left(a^{\prime}\right)}{b^{\prime}-a^{\prime}}=k$$

Deduce that there exists $c \in(a, b)$ with $f^{\prime}(c)=k$. [You may assume the Mean Value Theorem.]