Question

What are the points of inflection, if any, of `f(x) = 8 x^4−9 x^3 +9`?

Answer 1

`x=0,9/16`

Explanation:

The points of inflection occur when `f''(x)` switches sign.

`f(x)=8x^4-9x^3+9`
`f'(x)=32x^3-27x^2`
`f''(x)=96x^2-54x=6x(16x-9)`

The possible points of inflection occur when `f''(x)=0`.
This is when `x=0,9/16`.

Create a sign chart with the possible points of inflection.

`color(white)(xxxxxxxxxxx)0color(white)(xxxxxxxxxxx)9/16`
`larr----------------rarr`
`color(white)(xxxxxx)color(red)+color(white)(xxxxxxxx)color(red)-color(white)(xxxxxxxxxxxx)color(red)+`

Since the sign changes before and after `0` and `9/16`, they are both points of inflection.

graph{8x^4-9x^3+9 [-11.71, 16.77, -0.81, 13.43]}