= Solution
A <global minimizer> satisfies $I(\bar u)\leq I(v)$ for every admissible $v$.
A <point of convexity> $s$ of a differentiable scalar <function> is one where its tangent supports the entire graph: $g(q)\geq g(s)+g'(s)(q-s)$ for every $q\in\mathbb R$. The <Weierstrass necessary condition> for strong minimality, under its differentiability assumptions in the slope variable, is
$$
f(x,\bar u,q)-f(x,\bar u,\bar u')-f_v(x,\bar u,\bar u')(q-\bar u')\geq0\quad(q\in\mathbb R)
$$
for almost every $x$. In other words the reference slope is a <point of convexity> of the slope integrand. The later example is smooth, so these derivative-based conditions apply there.
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