On properties of univariate max functions at local maximizers

More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of a transfer function at maximizers. Their result extends easily to the statement that the maximum eigenvalue of a univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, at all local maximizers, a property that is useful in several applications that we describe. We also investigate whether this smoothness property extends to max functions more generally. We show that the pointwise maximum of a finite set of q-times continuously differentiable univariate functions must have zero derivative at a maximizer for q = 1, but arbitrarily close to the maximizer, the derivative may not be defined, even when q = 3 and the maximizer is isolated.