Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 1 / 10E / a / Solution

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then some $c\in (a,b)$ has $f\prime (c)=0$. A maximum or minimum is attained; unless $f$ is constant, one extremum lies inside, where the derivative vanishes.