The mean value theorem

Inside any interval there is always a tangent parallel to the secant joining the two ends. That is the mean value theorem¹. We confirm it for $f(x) = x^2$ on the interval $[1, 2]$.

The slope of the secant

Looking for a tangent of the same slope

Since $f'(x) = 2x$, solving $f'(c) = 3$ gives $c = \dfrac{3}{2}$, which lies inside the interval $1 < c < 2$.

The point of tangency is $\left( \dfrac{3}{2}, \dfrac{9}{4} \right)$ and the tangent is $y = 3x - \dfrac{9}{4}$. On the graph it appears as a line parallel to the secant $y = 3x - 2$.

What the theorem says

If $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then at least one $c$ between $a$ and $b$ satisfies the following.

At least one, so there is no objection to there being two or more.

Rolle's theorem

The case $f(a) = f(b)$ is a special one. The right side is then $0$, so some $c$ inside the interval has $f'(c) = 0$. That is Rolle's theorem².

The hypotheses are needed

Continuity alone is not enough without differentiability. Look at $f(x) = |x|$ on $[-1, 1]$: the secant has slope $0$, but there is no tangent of slope $0$ anywhere, because the function is not differentiable at $x = 0$.

What it is good for

The mean value theorem is the bridge between an average rate of change and a derivative at a point. Even the seemingly obvious fact that $f' > 0$ implies an increasing function is proved from it, by taking two points and writing $f(b) - f(a) = f'(c)(b - a) > 0$.

The parabola on the graph is $y = x^2$, the line through the two points is the secant, the line parallel to it is the tangent, and the large dots are $(1, 1)$, $(2, 4)$ and the point of tangency.

1. Mean value theorem, Wikipedia
2. Rolle's theorem, Wikipedia