= Bounded slope integrands force strong minima to be global
{title2=$I(u)=\int g(u'),\quad g\in C^1(\mathbb R)\text{ bounded}$}
Suppose the <Weierstrass necessary condition> applies to a strong local minimum of this fixed-endpoint integral. It requires the tangent to $g$ at $u'(x)$ to lie below the entire graph for almost every $x$. A line of nonzero slope is unbounded above in one direction, while $g$ is bounded; hence this supporting line has slope zero. Its contact value is then a <global minimum> of $g$. The resulting <function> minimizes the integral globally, since its integrand is almost everywhere equal to the global lower bound. Thus for a bounded differentiable slope-only integrand a strong local minimum can exist only at global energy level. Weak local minima can behave differently because their competitors have nearby derivatives.
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