An Odd Mean Value Theorem Problem
Solution 1:
Using Cauchy's Mean Value Theorem with functions $\frac{f(x)}x$ and $\frac1x$, we get $$ \frac{x_1f(x_2)-x_2f(x_1)}{x_1-x_2}=\frac{\frac{f(x_2)}{x_2}-\frac{f(x_1)}{x_1}}{\frac1{x_2}-\frac1{x_1}} =\frac{\frac{cf'(c)-f(c)}{c^2}}{-\frac1{c^2}}=f(c)-cf'(c) $$ for some $c\in[x_1,x_2]$. For this, I believe we may need that $0\not\in[x_1,x_2]$.