For the function, find all points of inflection or determine that no such points exist.
We need to use some quotient rule stuff here. First, the first derivative:
Usually we say not to bother squaring the denominator when we use the quotient rule. However, this denominator is only one term, so it looks better after we square it.
The second derivative has no roots and is undefined at x = 0. However, since x = 0 isn't in the domain of f to begin with, it can't be an inflection point. Therefore the function f (x) has no inflection points, which indeed it does not, it's concave up everywhere it is defined:
This might seem like a lot of work for no inflection points. However, knowing a graph has no inflection points is still a useful piece of information.