A real differentiable function is convex if and only if its derivative is monotonically increasing
Your friend is right.
From the previously solved exercise, you can show that for arbitrary $s<t$ in $(a,b)$, $\lim\limits_{u\to s+}\frac{f(u)-f(s)}{u-s}\leq\lim\limits_{v\to t-}\frac{f(t)-f(v)}{t-v}$, so $f'(s)\leq f'(t)$.