= Solution
\b[Zero is neither a weak nor a strong local minimizer.] For any nonzero smooth zero-trace $\varphi$, <Taylor expansion> gives
$$
I(\varepsilon\varphi)-I(0)=\varepsilon^2\left[-\frac12\int(\varphi+\varphi')^2+\frac18\int\varphi'^2+\int\varphi^2\right]+O(\varepsilon^4).
$$
The mixed <integral> $\int\varphi\varphi'=0$ by the endpoint values. Writing $D=\int\varphi'^2$ and $U=\int\varphi^2$, the quadratic coefficient is $-3D/8+U/2$. The elementary estimate $U\leq D/2$ on $[0,1]$ makes this at most $-D/8<0$. Thus arbitrarily small variations decrease the energy. The separate coefficient one on $u^2$ is retained in this calculation.
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