Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 11E / c

If $f\prime \ge 0$, the mean value theorem gives $f(y)-f(x)=f\prime (c)(y-x)\ge 0$. Conversely, if $f$ is increasing, every difference quotient with positive or negative increment is nonnegative; taking its limit gives $f\prime (x)\ge 0$.