The author consider the nonlinear second-order elliptic boundary value problem $Lu(x)=\lambda f(x,u(x))$, for $x$ in a regular, bounded domain $\Omega$, with boundary conditions $Bu(x)=0$ for $x\in\partial\Omega$, where $B$ is a linear boundary operator of Dirichlet, Neumann, or mixed type. It is assumed that for each $x\in\Omega$, $f(x,\cdot)$ is strictly positive, strictly increasing, and strictly convex on $(0,+\infty)$. Let $\Lambda$ be the set of positive real numbers $\lambda$ for which the problem posed has a positive solution (it is known that $\Lambda$ is an interval), and let $\lambda^\ast=\sup(\Lambda)$ be the ``critical parameter''. It is shown that Newton's method starting with a suitable subsolution yields a sequence of iterates which is not a monotonically increasing sequence if $\lambda>\lambda^\ast$ or $\lambda=\lambda^\ast\in\Lambda$, whereas for $0<\lambda<\lambda^\ast$ the sequence of iterates is monotonically increasing. This result is demonstrated for the abstract problem $u=\lambda Tu$, where $T$ is a forced, positive, isotone, convex, twice-differentiable compact operator on the positive cone of a partially ordered Banach space. The result is used to obtain quite precise upper and lower bounds on the value of the critical parameter $\lambda^\ast$ for five specific elliptic boundary value problems. The proofs use the general theory of such problems as presented, for example, by H. Amann and the reviewer [Indiana Univ. Math. J. {\bf 25} (1976), no. 3, 259--270; MR0417570 (54 \#5620)].