let f(x) be a diffrentiable function .if $\displaystyle f(0)=0,f(1)=1$,then the minimum value of $\displaystyle $\int_{0}^{1}( f'(x))^2dx$$

some one give me following hint

Hint: Integrate by parts:

$\displaystyle \int_0^1 (f'(x))^2\,\mathrm{d}x=\left.f(x)f'(x)\right\vert_ 0^1-\int_0^1f(x)f''(x)\,\mathrm{d}x$

Or equivalently, recall principle of least action/energy.[/quote]

sir i never heard about this principal .can you tell us how can we solve this question with the help of least action principal