# What are the points of inflection, if any, of f(x)= 9x^3- 5 x^2-2 ?

Jan 13, 2016

when $x = \frac{5}{27}$

#### Explanation:

A point of inflection occurs when the second derivative of a function switches sign (goes from positive to negative, or vice versa).

Find $f ' ' \left(x\right)$:

$f \left(x\right) = 9 {x}^{3} - 5 {x}^{2} - 2$
$f ' \left(x\right) = 27 {x}^{2} - 10 x$
$f ' ' \left(x\right) = 54 x - 10$

The second derivative could from positive to negative or negative to positive when $f ' ' \left(x\right) = 0$. Find those points:

$54 x - 10 = 0$

$x = \frac{10}{54} = \frac{5}{27}$

Check to make sure the second derivative actually changes sign around this point. As of now, it is just a possible point of inflection.

When $x < \frac{5}{27}$, we can test $f ' ' \left(0\right)$:

$f ' ' \left(0\right) = - 10 \text{ ... } < 0$

When $x > \frac{5}{27}$, we can test $f ' ' \left(1\right)$:

$f ' ' \left(1\right) = 44 \text{ ... } > 0$

Since the sign of the second derivative does change around $x = \frac{5}{27}$, it is a point of inflection.

We can check this graphically--the concavity should shift.

graph{(9x^3-5x^2-2) [-10, 10, -7.2, 2.8]}