A certain function $ f( x)$ defined for all $x$ has
$f^{''}(0)=0$, $f^{''}(x)>0$
for $x>0$, and $f^{''}(x)<0$ for $x<0$.
The number of critical points, number of
local maxima, number of local minima, and number of roots of $ f( x)$ are all
tabulated. Give all possible such tabulations.{({NOTE:} this problem
appeared on Dr. Hamrick's 408C exam in 1988.)}
Tags: critical-points, concavity, abstract-function, exam-UT, generalization, prediction, v--g, t1, c2
I absolutely enjoy watching students work this problem. It is helpful to have them draw a table with a column headings: # roots, # critical points, # local max., #local min. There are 14 tabulations, but don’t tell your students that. Part of the exercise is to have them convince you and themselves that there are no more than 14 (once they’ve gotten this far). Great qualitative problem!
– James Epperson 1998-08-09