Proposition: Existence and a conditional maximizing branch

Let $a(e,f)$ be continuous on a compact rectangle. For each $e$, at least one maximizing $f$ exists.

Suppose, near $(e_{\ast },f_{\ast })$, that $a$ is $C^2$, $f_{\ast }$ is the unique interior global maximizer,

Then, after restricting to a neighborhood of $e_{\ast }$, the maximizer is a $C^1$ branch satisfying

An increasing branch therefore requires a sign condition on $a_{ef}$. For a smooth stress surface $s$, stress decreases along this branch exactly when

The condition $s_f<0$ alone does not imply (eq:stressalongpeak).

Proof. Existence is the extreme-value theorem. The implicit-function theorem applied to $a_f=0$ gives the local branch and (eq:implicitpeak). Negative second derivative preserves a local maximum. Uniqueness of the original global maximum, compactness outside a small neighborhood, and continuity preserve its global status for nearby $e$. Equation (eq:stressalongpeak) is the chain rule.