# What are the points of inflection, if any, of f(x) =-3x^3 - 7x^2 + 3x?

Jun 5, 2017

The point of inflection is $= \left(- 0.778 , - 5.156\right)$#

#### Explanation:

We calculate the first and second derivatives

$f \left(x\right) = - 3 {x}^{3} - 7 {x}^{2} + 3 x$

$f ' \left(x\right) = - 9 {x}^{2} - 14 x + 3$

$f ' ' \left(x\right) = - 18 x - 14$

The point of inflection is when

$f ' ' \left(x\right) = 0$

$- 18 x - 14 = 0$, $\implies$, $x = - \frac{14}{18} = - \frac{7}{9}$

Therefore, the point of inflection is

$\left(- \frac{7}{9} , f \left(- \frac{7}{9}\right)\right) = \left(- 0.778 , - 5.156\right)$

We can build a chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$\left(- \infty , - \frac{7}{9}\right)$$\textcolor{w h i t e}{a a a a}$$\left(- \frac{7}{9} , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$f ' ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a}$$\cup$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\cap$
graph{(y-(-3x^3-7x^2+3x))((x+0.778)^2+(y+5.156)^2-0.01)=0 [-11.24, 6.54, -8.084, 0.805]}