Professor of Mathematics Robert Kantrowitz ’82 was a colloquium speaker at Mississippi State University earlier this month. His talk, titled “What does the condition f”’ ≥ 0 mean for f?,” centered around functions whose third derivatives do not change sign on an interval.
Kantrowitz said that while calculus informs us how the first and second derivatives of a function provide insight into the shape of its graph, it is less apparent what geometric information is captured by the third derivative.
“It turns out that functions whose third derivatives do not change sign on an interval always enjoy a predictable relationship with two quadratic polynomials that are closely associated with it,” he said.
Kantrowitz also discussed the application and history of this problem, which dates back to ballistics of the early 1900s.