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Proof of the Mean Value Theorem

If $f$ is a function that is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists some $c$ in $(a,b)$ where

Proof:

Let $A$ be the point $(a,f(a))$ and $B$ be the point $(b,f(b))$.

Note that the slope of the secant line to $f$ through $A$ and $B$ is $\displaystyle{\frac{f(b)-f(a)}{b-a}}$.

Combining this slope with the point $(a,f(a))$ gives us the equation of this secant line:

Let $F(x)$ share the magnitude of the vertical distance between a point $(x,f(x))$ on the graph of the function $f$ and the corresponding point on the secant line through $A$ and $B$, making $F$ positive when the graph of $f$ is above the secant, and negative otherwise.

More succinctly,

We intend to show that $F(x)$ satisfies the three hypotheses of Rolle's Theorem.

First, $F$ is continuous on $[a,b]$, being the difference of $f$ and a polynomial function, both of which are continuous there.

Second, $F$ is differentiable on $(a,b)$, for similar reasons. $F$ is the difference of $f$ and a polynomial function, both of which are differentiable there.

Lastly, note at $a$ and $b$,

Thus, the conditions of Rolle's Theorem are satisfied and there must exist some $c$ in $(a,b)$ where $F'(c) = 0$.

Consider the derivative of $F(x)$:

Thus, for some $c$ in $(a,b)$,

Equivalently, we have shown there exists some $c$ in $(a,b)$ where