Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 11E / ii

The mean value theorem states that a function continuous on $[a,b]$ and differentiable on $(a,b)$, where $a<b$, has a point $\xi \in (a,b)$ such thatTo prove it using Rolle's theorem, subtract the secant line:This function has the required continuity and differentiability, with $h(a)=h(b)=0$. Rolle's theorem gives $h^{\prime }(\xi )=0$, which is exactly the claimed equation. Thus a tangent slope somewhere in the interval equals the overall secant slope.