how do I solve this?

Question

A function f has a second derivative f''(x)=(x+2)x^2(x-1)^3(-4)^4. For which values of x does the function have inflection points.

Answer 1

Set the second derivative equal to zero to find the POSSIBLE points of inflection.

(x + 2)x²(x - 1)³(-4)⁴ = 0 so x = -2, x = 0, x = 1

As I mentioned, some of these may be points of inflection. To know, we need to determine if the concavity changes on each side of these. If it does, they will be points of inflection. Remember to find the concavity, we evaluate f ''(x) at values in the interval in question. So, I'll put them on a number line and I'll check concavity on each side like this:

The only values that have changes of concavity on each side are x = -2 and x = 1 so those are the Points of Inflection.