Question

Let f be a continuously differentiable real-valued function on [0.1] such that ${\int }_{1/3}^{2/3}f\left(x\right)dx=0$. find the minimum value of $\frac{{\int }_0^1{f}^{\prime }(x{)}^{2}dx}{\left({\int }_0^{1}f\right(x)dx{)}^2}$.

Answer 1

${\int }^{2/3}1/3f\left(x\right)dx=0$
different both side
f(x) = x where u = any courrent
f again differentiate
$f^{\prime }\left(x\right)=0$
$f^{\prime }(x{)}^2=0$
$\int f^{\prime }(x{)}^2=0$
so $\frac{{\int }_0^1{f}^{\prime }(x{)}^2}{{\int }_0^{1}f\left(x\right)dx{)}^2}=0$