Question

If f ''(x) = x3(x−1)3(x+5)2, find the x-values for all the points of inflection. (Enter your answers as a comma separated list.)

Answer 1

Since at the points of inflection of any function f(x), f '' (x) = 0
and you are given f '' (x) already in factored form, this is quite simple.
All you have to do it set each of the factors equal to zero and solve
f '' (x) = x^3(x−1)^3(x+5)^2
x^3 = 0 ---> x = 0
(x-1)^3 = 0
x-1 = 0 ----> x = 1
(x+5)^2 = 0 ---> x = -5

Answer in the way required.

Answer 2

so at inflection, f" is zero
I see a triple zero a x=0, triple zero at 1, and a double at x-5