The Mean Value Theorem

The derivative is a local object: $f^{\prime }(c)$ depends only on an arbitrarily small neighborhood of $c$. The mean value theorem carries that local information to global conclusions about $f$ across a whole interval. It rests on one lemma about extrema, then Rolle's theorem, then a single tilt of the picture.

Vanishing derivatives at interior extrema

Absolute maxima and minima are the highest and lowest values a function attains. A relative extremum is highest or lowest only among nearby points.

At an interior relative extremum a differentiable function must have zero slope. This is Fermat's stationary-point condition.

A point where $f^{\prime }(c)=0$ (or where $f^{\prime }$ fails to exist) is a critical point. Fermat's lemma says an interior extremum is always a critical point, which reduces optimization on an interval to a finite search.

The endpoints join the search because Fermat's lemma is silent there; an extremum at $a$ or $b$ need not be critical. The procedure is exhaustive precisely because a continuous $f$ on a closed bounded interval attains its extrema (the extreme value theorem), and an interior one must be critical.

Rolle's theorem

If a differentiable function returns to its starting height, it must turn around somewhere in between, and at the turning point the tangent is horizontal.

Differentiability on all of $(a,b)$ cannot be dropped. The function $|x|$ on $[-1,1]$ has equal endpoint values but no point where the derivative is zero, because of the corner at the origin where $f^{\prime }$ does not exist.

Reading Rolle's theorem backwards bounds how many roots a function can have: a root of $f^{\prime }$ sits between any two roots of $f$, so few roots of $f^{\prime }$ mean few roots of $f$.

The mean value theorem

Tilt Rolle's picture. Instead of equal endpoints, allow $f(a)\ne f(b)$ and compare the tangent slope against the slope of the secant joining the two endpoints.

The right-hand ratio is the slope of the secant through $(a,f(a))$ and $(b,f(b))$; the theorem asserts that the tangent at some interior $c$ is parallel to it. The name reflects that the secant slope is the average (mean) value of the derivative, and the average is actually achieved somewhere inside.

A more symmetric version replaces the linear comparison function by a second function $\phi$, giving Cauchy's mean value theorem: for continuous $f,\phi$ differentiable on $(a,b)$ there is a $c$ with

Taking $\phi (x)=x$ recovers the ordinary theorem. This form underlies both Taylor's theorem and L'Hôpital's rule.

From the sign of $f^{\prime }$ to the shape of $f$

The typical use of the mean value theorem is to remove a limit: it replaces the difference quotient by an exact derivative at some interior point, converting inequalities on $f^{\prime }$ into inequalities on $f$.

A vanishing derivative means constant. If $f^{\prime }(x)=0$ for all $x$ in an interval $I$, then $f$ is constant. For any $x<y$ in $I$ the theorem gives a $c$ with $f(y)-f(x)=f^{\prime }(c)(y-x)=0$, so $f(y)=f(x)$. This is the first differential equation solved in the subject: $f^{\prime }=0$ has only the constant solutions.³

The sign of $f^{\prime }$ controls monotonicity.

The strict converse fails: $f(x)=x^3$ is strictly increasing yet $f^{\prime }(0)=0$. A positive derivative is sufficient for strict growth but not necessary.

The first-derivative test. Combining the two directions, a sign change of the derivative locates an extremum. If $f^{\prime }\le 0$ on $(a,c)$ and $f^{\prime }\ge 0$ on $(c,b)$, then $f$ falls then rises, so $c$ is an absolute minimum on $(a,b)$; the reversed signs give a maximum. Restricting to $(c-\delta ,c+\delta )$ turns this into a test for relative extrema.⁵

Bounding differences: the mean value inequality

A bound on the derivative becomes a bound on how fast the function can change. If $|f^{\prime }|\le M$ throughout an interval, then for any $x,y$ the theorem gives a $c$ between them with

This is Lipschitz continuity with constant $M$: a differentiable function with bounded derivative cannot separate points faster than linearly.⁶ The converse direction also holds locally, so on an interval bounded derivative and Lipschitz coincide for differentiable functions.

The theorem also settles differentiability at an endpoint from the behavior of the derivative nearby.

Darboux's theorem

A derivative need not be continuous. Yet even a discontinuous derivative cannot jump over intermediate values; it has the intermediate value property regardless.

A discontinuous derivative genuinely exists.

This is why continuously differentiable functions get their own name and notation, $C^1(I)$: continuity of $f^{\prime }$ is a real extra assumption, not a free consequence of differentiability.