Putnam 1939 B6

13. Take either (i) or (ii). (i) Let $f(x)$ be defined for $a \leq x \leq b$. Assuming appropriate properties of continuity and derivability, prove for $a<x<b$ that $$\frac{\frac{f(x)-f(a)}{x-a}-\frac{f(b)-f(a)}{b-a}}{x-b}=\frac{1}{2} f^{\prime \prime}(\xi)$$ where $\boldsymbol{\xi}$ is some number between $\boldsymbol{a}$ and $\boldsymbol{b}$. (page 123) (ii) Calculate the mutual gravitational attraction of two uniform rods, each of mass $m$ and length $2 a$, placed parallel to one another and perpendicular to the line joining their centers at a distance $b$ apart.

In your answer let $a$ approach zero, and comment on the form of the result.