Rolle's theorem from attained extrema

Write $k=g(a)=g(b)$. If $g$ is constant, choose $c=(a+b)\mathrm{/}2\in (a,b)$; then $g^{\mathrm{\prime }}(c)=0$. Otherwise some value differs from $k$. If it is greater than $k$, [L2] gives an attained maximum greater than the endpoint values; if it is less, [L2] gives an attained minimum less than them. In either case an extremum occurs at an interior point $c$.

At an interior local maximum, positive difference quotients are nonpositive and negative difference quotients are nonnegative. Since the common two-sided limit exists by [L1], it must be both nonpositive and nonnegative, hence zero. The inequalities reverse at a local minimum and again force $g^{\mathrm{\prime }}(c)=0$. This proves the claim.