  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.
