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

Dec 23, 2015

$x = 0 , \frac{9}{16}$

#### Explanation:

The points of inflection occur when $f ' ' \left(x\right)$ switches sign.

$f \left(x\right) = 8 {x}^{4} - 9 {x}^{3} + 9$
$f ' \left(x\right) = 32 {x}^{3} - 27 {x}^{2}$
$f ' ' \left(x\right) = 96 {x}^{2} - 54 x = 6 x \left(16 x - 9\right)$

The possible points of inflection occur when $f ' ' \left(x\right) = 0$.
This is when $x = 0 , \frac{9}{16}$.

Create a sign chart with the possible points of inflection.

$\textcolor{w h i t e}{\times \times \times \times \times x} 0 \textcolor{w h i t e}{\times \times \times \times \times x} \frac{9}{16}$
$\leftarrow - - - - - - - - - - - - - - - - \rightarrow$
$\textcolor{w h i t e}{\times \times \times} \textcolor{red}{+} \textcolor{w h i t e}{\times \times \times \times} \textcolor{red}{-} \textcolor{w h i t e}{\times \times \times \times \times \times} \textcolor{red}{+}$

Since the sign changes before and after $0$ and $\frac{9}{16}$, they are both points of inflection.

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