Clamped second-variation inequality (${\int }_0^{\pi }{\eta }^2\le {\int }_0^{\pi }(\eta \prime {)}^2\le {\int }_0^{\pi }(\eta \prime \prime {)}^2$)

A perturbation preserving both values and slopes has $\eta =\eta \prime =0$ at the endpoints. Apply the Poincare 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.