# The Mean Value Theorem If the average rate of change of $f(x)$ between $x=a$ and $b$ is $s$, then there must exist at least one point $c$ between $a$ and $b$, such that $f'(c) = s$. The mean value theorem is applicable to all [[Smooth Functions]]. ## Proof The proof of the mean value theorem relies on [[Rolle's Theorem]]. Per Rolle, if we can find a function $g$ such that $g(a) = g(b) = 0$, and such that $g'(a) = f'(a) - s$, then we will have proven MVT because per Rolle there must exist at least one point $c$ such that $g'(c) = 0$. At that point $0 = f'(a) - s$, that is $f'(a) = s$. Recall that $s = \frac {f(b)-f(a)} {b-a}$. $g(x)$ can be defined as $g(x) = f(x) - f(a) - s (x-a)$ $g(a) = f(a) - f(a) - s(a-a) = 0$ $\begin{align}g(b) &= f(b)-f(a)-s(b-a) \\ &= f(b)-f(a) - (f(b) - f(a))\\ &=0\end{align}$ $ g'(k) = f'(k) - \frac d {dk} [f(a)] - s + \frac d {dk} sa $ The second and fourth terms of the above are constant and so go to 0, leaving just $g'(k) = f'(k) - s$.