For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = sin *x*

Answer

*f* (*x*) = sin *x*

The first derivative is *f *'(*x*) = cos *x* and the second derivative is

*f *"(*x*) = -sin *x*.

The second derivative is never undefined, and is zero at nπ for every integer *n*. Therefore our inflection points are *n*π for every integer *n*. If we look at the graph, this makes sense. We can think of the graph of sin *x* as a series of upside down and right-side up bowls, and an inflection point occurs between every two bowls: