# What are the points of inflection of f(x)= x^3 + 5x^2 + 4x - 3?

Oct 25, 2016

There is only one point of inflection at $\left(- 1.667 , - 0.407\right)$

#### Explanation:

To determine points of inflexion, we look when $f ' ' \left(x\right) = 0$

$f \left(x\right) = {x}^{3} + 5 {x}^{2} + 4 x - 3$

$f ' \left(x\right) = 3 {x}^{2} + 10 x + 4$

$f ' ' \left(x\right) = 6 x + 10$

$f ' ' \left(x\right) = 0 \implies 6 x + 10 = 0$
$x = - \frac{10}{3} / 6 = - \frac{5}{3} = - 1.667$

$f \left(- \frac{5}{3}\right) = {\left(- 1.667\right)}^{3} + 5 {\left(- 1.667\right)}^{2} + 4 \left(- 1.667\right) - 3 = - 0.407$

graph{x^3+5x^2+4x-3 [-10.915, 9.08, -5.74, 4.26]}