The authors show that the maximum of any component of the gradient of a minimum of the integral functional \[ I(u) =\int_\Omega [f(Du) + g(u)] dx \] must occur on the boundary of the domain \(\Omega\) provided the functional \(I\) is strictly convex. No further regularity (or growth) conditions are assumed. Such results are well known (see, for example, [\textit{N. S. Trudinger}, Math. Z. 109, 211-216 (1969; Zbl 0174.15801)] or [\textit{M. Giaquinta} and \textit{L. Pepe}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 25, 481-507 (1971; Zbl 0283.49032)]) when \(f\) and \(g\) are sufficiently smooth, so the main concern is in showing that the argument in these references can be applied in the more general situation. The main focus in the present paper is on a comparison principle for what the authors call subminima and supermaxima (which are the analogs of subsolutions and supersolutions of elliptic equations). The authors also consider some applications to constrained problems.