Question

What are the inflection points of f(x)?

`f(x)=1.5x^5-5x^4-20x^3+120x^2-2x-6`

Answer 1

`(-2, 510)` is the only point of inflection.

Explanation:

We must first find the second derivative.

`f'(x) = 7.5x^4 - 20x^3 - 60x^2 + 240x - 2`
`f''(x) = 30x^3 - 60x^2 - 120x + 240`

Inflection points occur when `f''(x) = 0`.

`0 = 30x^3 - 60x^2 - 120x + 240`

Thankfully we can readily solve this by factoring.

`0 = 30x^2(x - 2) -120(x - 2)`

`0 = (30x^2 - 120)(x- 2)`

`x = 2 or x^2 = 4`

`x = +-2`

But before declaring these, we must test the sign of the second derivative between these points. An inflection point only occurs if the sign changes.

Consider the following table:

As you can see, the second derivative does equal `0` at `x = 2`, but it does not goes from positive to negative. Instead it stays positive. Therefore, we must reject `x = 2`.

However, `x= -2` does have a sign change, therefore it's acceptable.

Hopefully this helps!