Let $f$ be a continuous function on $[a,b]$, and continuously differentiable on $(a,b)$. Assume that $f'$ is bounded on $(a,b)$ and $\sup_{(a,b)}|f'(x)| = K$.
We'll also denote the upper and lower Darboux sums with respect to a partition $P$ of $[a,b]$ by $U(f, P)$ and $L(f, P)$ respectively. We need to show that for every partition of $[a,b]$ it holds that: $0 \le U(f, P) - L(f, P) \le K(b-a)\Delta(P)$
Where $\Delta(P)$ represents the diameter of $P$.
I tried using Lagrange's mean value thm, or uniform continuity but with no success.
I would appreciate your help with this question.
Thanks.