# What are the points of inflection, if any, of f(x)= 7x^5 - 12x^3 + x ?

Nov 7, 2016

Points of inflection are at
$x = 0$
$x = \pm \sqrt{\frac{18}{35}} \approx \pm .71714$

#### Explanation:

To find points of inflection, set $f ' ' \left(x\right)$ equal to zero. To do this, first find f'(x) using the power rule:

$f \left(x\right) = 7 {x}^{5} - 12 {x}^{3} + x$

$f ' \left(x\right) = 35 {x}^{4} - 36 {x}^{2} + 1$

$f ' ' \left(x\right) = 140 {x}^{3} - 72 x$

$0 = 140 {x}^{3} - 72 x$
$0 = \left(4 x\right) \left(35 {x}^{2} - 18\right)$
$x = 0$
$x = \pm \sqrt{\frac{18}{35}} \approx \pm .717137$