The Sturm-Liouville problem \(-f''+qf=\lambda rf\), \(f(-1)=f(1)=0\) is studied on \([-1,1]\) with \(r,q\in L^1[-1,1]\) real, \(q>-\pi^2/4\) a.e. when \(r\) can change its sign. It is proved that the critical point of the operator \(Af:=\frac{1}{r} (-f''+qf)\) is regular under the conditions \(r < 0\) a.e. on \([-1,0)\), \(r>0\) a.e. on \((0,1]\) and \(r(x)=x^\nu g(x)\) on \((0,\varepsilon)\), \(\varepsilon\in (0,1]\), \(\nu>-1\), \(g\in C^1[0,\varepsilon]\), \(g(0)\neq 0\).