 Solution

Assume $c\le 0$, bounded coefficients and uniform ellipticity near the boundary point, an interior tangent ball there, and enough regularity for the boundary normal derivative. The Hopf boundary point lemma says that a nonnegative maximum at $x_0$ with strictly smaller values inside the tangent ball satisfiesfor the outward unit normal. If $c=0$, the maximum can have either sign. Without existence of a classical derivative, one states a strict one-sided derivative bound.

The PDF also asks to deduce the strong maximum principle for elliptic operators; that request is missing from the TeX transcription. Suppose $v$ attains a nonnegative global maximum $M$ inside a connected domain but is not constant. A ball in $\{v<M\}$ can be chosen relatively compact in the domain and tangent to $\{v=M\}$ at an interior point $z$: use the distance from a nearby point to the closed maximum set. The Hopf boundary point lemma on that ball gives a nonzero outward derivative at $z$, contradicting $Dv(z)=0$ at an interior maximum. Therefore $v$ is constant. When $c=0$, the same proof works for a maximum of either sign.