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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 <jats:italic>q<\/jats:italic>-times continuously differentiable univariate functions must have zero derivative at a maximizer for <jats:inline-formula><jats:alternatives><jats:tex-math>$$q=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, but arbitrarily close to the maximizer, the derivative may not be defined, even when <jats:inline-formula><jats:alternatives><jats:tex-math>$$q=3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the maximizer is isolated.<\/jats:p>","DOI":"10.1007\/s11590-022-01872-y","type":"journal-article","created":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T13:05:38Z","timestamp":1648472738000},"page":"2527-2541","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On properties of univariate max functions at local maximizers"],"prefix":"10.1007","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8426-0242","authenticated-orcid":false,"given":"Tim","family":"Mitchell","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6563-6371","authenticated-orcid":false,"given":"Michael L.","family":"Overton","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,28]]},"reference":[{"issue":"5","key":"1872_CR1","doi-asserted-by":"publisher","first-page":"A3609","DOI":"10.1137\/17M1137966","volume":"40","author":"P Benner","year":"2018","unstructured":"Benner, P., Mitchell, T.: Faster and more accurate computation of the $${\\cal{H}}_\\infty $$ norm via optimization. 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