What function f(x) meets the requirements below and gives the smallest arc length possible? f(x)=0, f(1)=0 f(x) or = 0 0

Question

What function f(x) meets the requirements below and gives the smallest arc length possible?
f(x)=0, f(1)=0
f(x)> or = 0
0< or = x <or = 1 (x is between 0 and 1) 

(I think it's some type on minimization problem? But I don't know what to do..)

confluxepic · Answer

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anonymous · Answer

The arc length $L$ of a function $f(x)$ is given by the integral formula,
$$L=\int_CdS=\int_0^1\sqrt{1+[f'(x)]^2}~dx$$
Regardless of the sign of $f'(x)$, since it is being squared, the minimum possible value is $f'(x)=0$, which means $f(x)$ is a constant function.

Only one constant function satisfies $f(0)=f(1)=0$.