Question Number 105742 by ZiYangLee last updated on 31/Jul/20
![Let a differentiable function f:R→R satisfies ∣f′(x)∣≤1 for all x∈[0,2] and f(0)=f(2)=1 Prove that 1≤∫_0 ^2 f(x)dx≤3](Q105742.png)
$$\mathrm{Let}\:\mathrm{a}\:\mathrm{differentiable}\:\mathrm{function}\:\mathrm{f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$$\mathrm{satisfies}\:\mid\mathrm{f}'\left(\mathrm{x}\right)\mid\leqslant\mathrm{1}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}\in\left[\mathrm{0},\mathrm{2}\right]\:\mathrm{and} \\ $$$$\mathrm{f}\left(\mathrm{0}\right)=\mathrm{f}\left(\mathrm{2}\right)=\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{1}\leqslant\int_{\mathrm{0}} ^{\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\leqslant\mathrm{3}\: \\ $$