= Solution
The result is weaker than the pointwise <null energy condition> used in the elementary proof of the <second law of black-hole mechanics>: $T_{ab}U^aU^b$ may be negative over a finite interval, so the expansion and area can have local behavior that the pointwise argument does not control. Nevertheless, the <averaged null energy condition for a nonminimally coupled scalar> supplies the integrated positivity required by strengthened global focusing and area theorems when their completeness, endpoint, and genericity hypotheses hold. Thus the calculation supports a suitable global second law for $\xi<0$, but the displayed averaged inequality alone does not prove pointwise area monotonicity without those additional assumptions.
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