For the function, find all points of inflection or determine that no such points exist.
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 = e3/2 ≅ 4.48.
The numbers x = 4 and x = 5 are nice ones to use.
Since the sign of f" does change, x = e3/2 is indeed an IP.
From the graph, this is believable: