Sharp area-cost bound for a barrier with fixed endpoints (${\int }_0^1{y}^{\prime 2}dx\ge 12({\int }_0^{1}ydx{)}^2$)

For $y(0)=y(1)=0$, write $A={\int }_0^{1}ydx$ and $w=x(1-x)$. Integration by parts gives $2A={\int }_0^1{y}^{\prime }{w}^{\prime }dx$, while ${\int }_0^1{w}^{\prime 2}dx=1/3$. The Cauchy-Schwarz inequality therefore gives the displayed sharp bound, with equality exactly when $y=Kw$. A prescribed positive budget $B={\int }_0^1(y+y^{\prime 2})dx$ consequently givesThis proves the unique global optimum, rather than only stationarity from a Lagrange multiplier.