The mean value theorem (equality)

The mean value equality

Mean value theorem. Let $f$ be continuous on $[a \,;\, b]$ and differentiable on $\left]a \,;\, b\right[$, with $a < b$. There exists at least one $c$ in $\left]a \,;\, b\right[$ such that

$$f(b) - f(a) = f'(c)\,(b - a)$$

In other words, the slope of the chord joining the two ends of the curve is attained by at least one tangent:

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Rolle's theorem is the special case $f(a) = f(b)$, and conversely the mean value theorem is proved from Rolle.

Proof from Rolle

The idea is to subtract the chord in order to reduce to Rolle's case. For $x$ in $[a \,;\, b]$, set:

$$g(x) = f(x) - f(a) - \frac{f(b) - f(a)}{b - a}\,(x - a)$$

The function $g$ is continuous on $[a \,;\, b]$ and differentiable on $\left]a \,;\, b\right[$, as the sum of $f$ and an affine function. We compute

$$g(a) = 0 \qquad \text{and} \qquad g(b) = f(b) - f(a) - \frac{f(b) - f(a)}{b - a}\,(b - a) = 0$$

Rolle's theorem therefore applies to $g$: there exists $c$ in $\left]a \,;\, b\right[$ such that $g'(c) = 0$. Now

$$g'(x) = f'(x) - \frac{f(b) - f(a)}{b - a}$$

hence $f'(c) = \dfrac{f(b) - f(a)}{b - a}$, which is the desired conclusion.

Cauchy's version

By replacing the affine function with a second function $g$, the same scheme gives Cauchy's generalization: if $f$ and $g$ are continuous on $[a \,;\, b]$ and differentiable on the open interval, there exists $c$ such that

$$\left(f(b) - f(a)\right)\,g'(c) = \left(g(b) - g(a)\right)\,f'(c)$$

It is this version that underlies L'Hôpital's rule, and we recover the mean value theorem by taking $g(x) = x$.

Classic pitfall

The theorem is false for complex-valued or vector-valued functions. Consider $f(t) = e^{it}$ on $[0 \,;\, 2\pi]$. We have $f(2\pi) - f(0) = 0$, whereas

$$f'(t) = i\,e^{it} \qquad \text{satisfies} \qquad \left\lvert f'(t) \right\rvert = 1 \neq 0 \quad \text{for every } t$$

No $c$ works. What survives in this setting is the mean value inequality, the subject of the next lesson — and that is precisely why it is the tool actually used in analysis.