Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 11E (Analysis I)

Rolle's theorem states that, if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, $a<b$, and $f(a)=f(b)$, thenThe endpoint equality and both regularity hypotheses are part of the assertion.

The mean value theorem states that a function continuous on $[a,b]$ and differentiable on $(a,b)$, where $a<b$, has a point $\xi \in (a,b)$ such thatTo prove it using Rolle's theorem, subtract the secant line:This function has the required continuity and differentiability, with $h(a)=h(b)=0$. Rolle's theorem gives $h^{\prime }(\xi )=0$, which is exactly the claimed equation. Thus a tangent slope somewhere in the interval equals the overall secant slope.

First $g(b)\ne g(a)$: otherwise Rolle's theorem would produce a point with $g^{\prime }=0$, contrary to the hypothesis. DefineIt is continuous on $[a,b]$, differentiable inside, and vanishes at both endpoints. Rolle's theorem gives a $\xi \in (a,b)$ withThe two denominators are nonzero, so this proves Cauchy's mean value theorem:When $f(a)=g(a)=0$, apply that theorem on $[a,x]$ for each $a<x\le b$. Rolle's theorem again ensures $g(x)\ne 0$, and some ${\xi }_x\in (a,x)$ satisfies $f(x)/g(x)=f^{\prime }({\xi }_x)/g^{\prime }({\xi }_x)$. As $x\downarrow a$, the trapped point ${\xi }_x$ also tends to $a$. ThereforeThis proves the stated endpoint case of L'Hôpital's rule. It does not assume continuity of the derivatives; only the indicated derivative-ratio limit is used. The same trapping argument also works for an extended infinite limit if that convention is allowed.

Set $F(h)=f(a+h)+f(a-h)-2f(a)$ and $G(h)=h^2$. Both vanish at zero. For $h\ne 0$, their derivative ratio isTwice differentiability at $a$ makes both quotients tend to $f^{\prime \prime }(a)$. Apply the endpoint L'Hôpital's rule proved in part (iii) for $h\downarrow 0$. For the other side use the reflected positive variable $t=-h$; $F$ and $G$ are even, so their quotient is unchanged. Thus the two-sided limit isThis is the second-order central difference limit. The proof only uses differentiability of $f^{\prime }$ at $a$ and differentiability nearby; it does not incorrectly assume $f^{\prime \prime }$ is continuous.