Question Number 125621 by ZiYangLee last updated on 12/Dec/20
![Consider a continuously differentiable function f:[0,1]→R such that f(0)=0 and f(1)=1. Find the minimum value of ∫_0 ^1 (f′(x))^2 (√(1+x^2 )) dx](https://www.tinkutara.com/question/Q125621.png)
$$\mathrm{Consider}\:\mathrm{a}\:\mathrm{continuously}\:\mathrm{differentiable} \\ $$$$\mathrm{function}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that}\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\mathrm{and}\:{f}\left(\mathrm{1}\right)=\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}'\left({x}\right)\right)^{\mathrm{2}} \sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx} \\ $$