The following extremum problem is studied: ∫ ₀ ¹(y ^″(t)) ^p dt / ∫ ₀ ¹(y ^′(t)) ^q dt → min over all y, with y(0) = y(1) = 0 and y′(0) = y′(1) =0, which leads to the integral ∫ℝ(max(0, 1 + μx - |x| ^q)) ^1/p′ dx and yields exact estimates for the eigenvalues of differential operators in the generalized Lagrange problem on the stability of a column.