Question

What are the points of inflection, if any, of #f(x)=x^4/(x^3+6 #?

Answer 1

`(root(3)12, (2root(3)12)/3)`

Explanation:

`f''(x) = -(36x^2(x^3-12))/(x^3+6)^3`

An inflection point of `f` is a point on the graph of `f` at which the concavity changes.

`f''` changes signs at `x=root(3)12` and at `-root(3)6`

`-root(3)6` is not in the domain of `f`, so there is no point on the graph pf `f` with `x = -root(3)6`.

`root(3)12` is in the domain of `f`, and `f(root(3)12) = (2root(3)12)/3`