Let $n\geq 2$ and $f\in \mathcal{C}^2([|1,n|],\mathbb{R})$ such that $|f’|$ has a maximum strictly less than $1$ and $f’’$ has a minimum strictly positive. Show that the number of integer coordinates in the graph of $f$ is at most $1+2n^{\frac{2}{3}}$
This exercise appears in a worksheet and despite drawing various things, i can’t figure out how to prove it. Maybe by induction on $n$ but we might need assumptions on $f(n)$ and $f’(n)$…