State and prove the Intermediate Value Theorem.

State the Mean Value Theorem.

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

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

and

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

or otherwise, show that there is a subinterval $[\alpha, \beta] \subseteq[a, b]$ such that

$$\frac{g(\beta)-g(\alpha)}{\beta-\alpha}=k$$

Deduce that there exists $c \in(a, b)$ with $g^{\prime}(c)=k$.