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

May 6, 2018

The point of inflection is $= \left(0.269 , - 2.051\right)$

#### Explanation:

Calculate the first and second derivatives

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

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

$f ' ' \left(x\right) = 20 {x}^{3} + 6 x - 2$

The points of inflection are when

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

$20 {x}^{3} + 6 x - 2 = 0$

$\iff$, $10 {x}^{3} + 3 x - 1 = 0$

The solution of this equation is obtained graphically

$x = 0.269$

The point of inflection is $= \left(0.269 , - 2.051\right)$

graph{(y-(x^5+x^3-x^2-2))(y-(10x^3+6x-2))=0 [-2.712, 4.217, -4.075, -0.61]}