Question

Do all functions have points of inflection?

Answer 1

No. Consider `f(x)=x` - this function's concavity does not change throughout the entire run of the function.

All polynomials with odd degree of 3 or higher have points of inflection, and some polynomials of even degree (again, higher than 3) have them. The best way to determine if a function has a point of inflection is to look at its second derivative - if the second derivative can equal zero, the original function has a point of inflection.