Question

If f(0)=f(1) = 0 and f'' exists

Answer 1

If f(0)=f(1) = 0 and f'' exists, show that

\[\int\limits_{0}^{1}f''(x)f(x)dx =-\int\limits_{0}^{1}[f'(x)]^2dx\]

Answer 2

Do Integration by parts
\[
\int_0^1 u dv =\left [ uv \right]_0^1- \int_0^1 v du

\]

Answer 3

Put \[
dv = f''(x) dx;\quad u=f(x) \\
v= f' ;\quad du =f'(x) dx\\
\int\limits_{0}^{1}f''(x)f(x)dx = \left[ f(x) f'(x) \right]_0^1-\int\limits_{0}^{1} f'(x) f'(x)dx\\
\int\limits_{0}^{1}f''(x)f(x)dx =0-\int\limits_{0}^{1} f'(x) f'(x)dx\\
\int\limits_{0}^{1}f''(x)f(x)dx = -\int\limits_{0}^{1} [f'(x)]^2 dx
\]

Answer 4

Nice one ^.^