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

Answer

We use the quotient rule a couple of times to find the first two derivatives:

The second derivative is undefined when *x* = 0, but that's not in the domain of *f* anyway so it doesn't count. The second derivative is 0 when

To confirm that this is an inflection point, we need to check what the sign of *f*" does as we cross *x* = *e*^{3/2} ≅ 4.48.

The numbers *x* = 4 and *x* = 5 are nice ones to use.

Since the sign of *f*" does change, *x* = *e*^{3/2} is indeed an IP.

From the graph, this is believable: