On a family of extremum problems and the properties of an integral

The following extremum problem is studied:
$$ \int _0^1\bigl(y''(t)\bigr)^p\,dt\bigg/ \int _0^1\bigl(y'(t)\bigr)^q\,dt \to\min $$
over all $y$, with $y(0)=y(1)=0$ and $y'(0)=y'(1)=0$, which leads to the integral
$$ \int_{\mathbb R}\bigl(\max(0,1+\mu x-|x|^q)\bigr)^{1/p'}\,dx $$
and yields exact estimates for the eigenvalues of differential operators in the generalized Lagrange problem on the stability of a column.