Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 6 / b / Solution

Use the local linear approximation from part (a), with $L=f^{\prime }(c)$. Since $x_n\le c\le y_n$ and $x_n\ne y_n$, we have $y_n-x_n>0$, and subtraction givesThe coefficients $(y_n-c)/(y_n-x_n)$ and $(c-x_n)/(y_n-x_n)$ are nonnegative and sum to one. ThereforeHere $ϵ(c)=0$ handles any endpoint of a secant that happens to equal $c$. Thus secants straddling a differentiability point satisfy