Shape of graph polynomial to the fifth degree

[Iclaudius]

Hello my friends,

I have this problem and would appreciate someones help:


Determine all intervals where the following function is increasing or decreasing.

F(x) = -x^(5)+(5/2)x^(4)+(40/3)x^(3)+5

Solution
To determine if the function is increasing or decreasing we will need the derivative.

F'(x) = -5x^(4)+10x^(3)+40x^(2)

factored
F'(x) = -5x^(2) (x-4)(x+2)

Ok so here is where i have difficulty, i know x = 0, x = 4, and x = -2 however I do not understand why x = 0.
I understand why x = 4, and x = -2 - from solving simple quadratic at y = 0 to identify where this function is not changing, however where does someone get the x = o from?

I have been looking at some online resources and they have not provided adequate explanation - this particular problem is from http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtI.aspx

oh btw i apologize for the messy math notation
Thanks in advance,
Claudius

 

[SteamKing]

Well, you do have two linear factors [ (x-4) and (x+2) ] multiplied by -5x^2.
Don't you think if you set x = 0, then f'(x) = 0 then?

 

[Iclaudius]

Ah yes :D, thank you!