Chapter 6.4, Problem 20ES

Let $y=f\left(x\right)$ be a smooth curve on the closed interval $\left[a,b\right]$ . Prove that if $m$ and $M$ are nonnegative numbers such that $m\le \left|f^{\prime }\left(x\right)\right|\le M$ for all $x$ in $\left[a,b\right]$ , then the arc length $L$ of $y=f\left(x\right)$ over the interval $\left[a,b\right]$ satisfies the inequalities

$\left(b-a\right)\sqrt{1+m^2}\le L\le \left(b-a\right)\sqrt{1+M^2}$