 Solution

SetIts endpoint values are equal. Rolle theorem therefore supplies $\xi \in (a,b)$ withIf $g^{\prime }(\xi )=0$, then (1) and $g(b)\ne g(a)$ force $f^{\prime }(\xi )=0$, contrary to the hypothesis. Hence $g^{\prime }(\xi )\ne 0$, and division in (1) gives

The condition is necessary. On $[0,2\pi ]$, letHere $g(2\pi )-g(0)=\pi /2$ while $f(2\pi )-f(0)=0$, so the endpoint ratio is zero. ButWhere $g^{\prime }(x)\ne 0$, their ratio is $1/(1+\frac{1}{2}\cos x)$ and is never zero. At the remaining points both derivatives vanish, so the derivative ratio is undefined. Thus the conclusion fails when simultaneous zeros are allowed.