Question

How do you find the inflection points of `f(x)=3x^5-5x^4-40x^3+120x^2`?

Answer 1

The inflection points are `x=-2,1,2`

Explanation:

To find the inflection points you need to perform the second derivative test. Since this is a polynomial we use the power rule to differentiate the equation, `nx^(n-1)`.
We get the first `d/dx` which is:

`f'=15x^4-20x^3-120x^2+240x`

Followed by the second `d/dx` which is:

`f''=60x^3-60x^2-240x+240`

Now we factor you should get:

`60x^2(x-1)-240(x-1)`

`(60x^2-240)(x-1)`

Now set the factors equal to zero:

`60x^2-240=0` and `x-1=0`

Solve them and you should get:

`x=+-2, 1`

If you wish to find out where the exact inflection points occur plug in the three values into the original equation :)
You would do `f(-2), f(1), f(2)`.