How one can show that $\int _0 ^1 ((h')^2-h^2)dx \ge0 $ for all $h\in C^1[0,1]$ and $h(0) = 0$? How one can show that $\int _0 ^1 ((h')^2-h^2)dx \ge0 $ for all $h\in C^1[0,1]$ and $h(0) = 0$?                                                   

One can't, because it is false. Consider $h(x) = 1$, $h'(x) = 0$, $\int_0^1 ((h'(x))^2 - (h(x))^2) d x = -1$.
