= Clamped second-variation inequality
{title2=$\int_0^\pi\eta^2\le\int_0^\pi(\eta\prime)^2\le\int_0^\pi(\eta\prime\prime)^2$}
A perturbation preserving both values and slopes has $\eta=\eta\prime=0$ at the endpoints. Apply the <Poincaré inequality> to $\eta$ and then to $\eta\prime$. The resulting inequality makes a quadratic <second variation> proportional to $\int(\eta^2-(\eta\prime\prime)^2)$ nonpositive. If a functional is quadratic, its exact expansion about a stationary function then proves a global maximum.
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