If n is a positive integer, how many times does the function f(x) = x^2 + 5cos(x) change concavity in the interval 0 is less than or equal to x which is less than or equal to 2*pi*n?
It changes concavity twice in the interval $\displaystyle [0,2 \pi]$, and as the second derivative is periodic with period $\displaystyle 2 \pi$ in any of the intervals $\displaystyle [2 n \pi, 2 (n+1) \pi]$, and most closed intervals of length $\displaystyle 2 \pi$ (the exeptions are intervals with a zero of $\displaystyle 2-5 \cos(x)$ at an end point.
RonL