
Notice that both conditions on $f$ are necessary. Without either one, the statement is false!
Though the theorem seems logical, we cannot be sure that it is always true without a proof.
The Mean Value Theorem is a generalization of Rolle's Theorem: We now let $f(a)$ and $f(b)$ have values other than $0$ and look at the secant line through $(a,f(a))$ and $(b,f(b))$. We expect that somewhere between $a$ and $b$ there is a point $c$ where the tangent is parallel to this secant.
That is, the slopes of these two lines are equal. This is formalized in the Mean Value Theorem.
Here, $f'(c)$ is the slope of the tangent at $c$, while $\displaystyle \frac{f(b)f(a)}{ba}$ is the slope of the secant through $a$ and $b$. Intuitively, we see that if we translate the secant line in the figure upwards, it will eventually just touch the curve at the single point $c$ and will be tangent at $c$. However, basing conclusions on a single example can be disastrous, so we need a proof.
Consequences of the Mean Value TheoremThe Mean Value Theorem is behind many of the important results in calculus. The following statements, in which we assume $f$ is differentiable on an open interval $I$, are consequences of the Mean Value Theorem:
Key Concepts
Let $f$ be differentiable on $(a,b)$ and continuous on $[a,b]$. Then there is at least one point $c$ in $(a,b)$ for which $$f'(c) = \frac{f(b)f(a)}{ba}.$$ 