= Parabolic obstacle under a taut string
{title2=$w=\begin{cases}h-x^2,&|x|\leq s,\\2s(1-|x|),&s\leq|x|\leq1,\end{cases}\quad s=1-\sqrt{1-h}$}
For a string of positive tension $T$ with fixed endpoints $w(\pm1)=0$, zero transverse loading and obstacle $f=h-x^2$, $0\leq h\leq1$, equilibrium minimizes $T\int_{-1}^1(w')^2/2\,dx$ among functions above $f$. On a free interval $w''=0$, while in contact $w=f$. Smooth tangency of the right line through $(1,0)$ to the parabola at $x=s$ requires $h-s^2=2s(1-s)$, yielding the displayed root in $[0,1]$.
The two pieces join with equal value and slope. On the free portions $w-f=(|x|-s)^2\geq0$; in contact $-Tw''=2T$ is the nonnegative reaction. These facts verify the <unilateral contact complementarity>[complementarity] conditions. The associated <variational inequality> and the exact energy difference prove global minimality; strict convexity with fixed endpoints proves uniqueness. At $h=0$ contact reduces to one point; at $h=1$ it fills the whole interval.
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