# What are the inflection points for x^3 + 5x^2 + 4x - 3?

May 25, 2015

An inflection point is defined when a function is changing from concave to convex (or vice-versa), that is, changing its concavity.

Thus, the second derivative:

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {x}^{2} + 10 x + 4$

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 6 x + 10$

Equaling the second derivative to zero (because at the inflection point, the slope's not changing, it's the very point where concavity changes, thus being of derivative equal to zero:

$6 x + 10 = 0$
$x = - \frac{10}{6} = - \frac{5}{3} \cong - 1.67$

Substituting this $x$ coordinate in the original function, in order to get the $y$ coordinate:

$y = {\left(- \frac{5}{3}\right)}^{3} + 5 {\left(- \frac{5}{3}\right)}^{2} + 4 \left(- \frac{5}{3}\right) - 3$
$y = \left(- \frac{125}{27}\right) + \left(\frac{125}{9}\right) - \frac{20}{3} - 3$
$y = \frac{- 125 + 375 - 180 - 81}{27}$
$y = - \frac{11}{9} \cong - 1.22$

So, you inflection point is $\left(- 1.67 , - 1.22\right)$

graph{x^3+5x^2+4x-3 [-10, 10, -5, 5]}