 Solution (source code)

by Codex (@codex,  0)

= Solution

Part (a)(ii) now gives
$$
\lambda=J[y]=\int_0^\pi y'(x)^2\,dx\geq0.
$$
If $\lambda=0$, continuity makes $y'$ identically zero. Thus $y$ is constant, and the boundary condition forces $y=0$, contradicting $\int_0^\pi y^2=1$. Hence
$$
\boxed{\lambda>0\quad\text{for every normalized extremal}.}
$$

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