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In Section 4.2, we have discussed Rolle's Theorem, which says that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with $f(a) = f(b)$, then there exists a point $c$ between $a$ and $b$ so that $f'(c) = 0$. The point of this workshop problem is to use this to prove something potentially more useful: the Mean Value Theorem.
Let $g$ be any function that is continuous on $[a,b]$ and differentiable on $(a,b)$.
\begin{enumerate}
    \item The only part we are missing is the equality of the endpoints. Let $h_1(x) = g(x) - g(a)$. What is $h_1(a)$?
    \item Now, let $h(x) = g(x) - g(a) - \frac{(x-a)}{(b-a)}g(b)$. What is $h(a)$? What is $h(b)$?
    \item Based on what Rolle's Theorem says, we now know that it applies to $h$. What does this mean in terms of $h$ and some point $c$ between $a$ and $b$?
    \item Use the definition of $h$, take the derivative, and use the result from Rolle's Theorem to get the following result: There exists a $c$ between $a$ and $b$ so that
    \[ g'(c) = \frac{g(b) - g(a)}{b-a} \]
\end{enumerate}
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