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

Jul 30, 2018

The function has three Inflection points at
$x = - 0.672 , x = - 0.176 , x = 0.847$.

#### Explanation:

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

$f ' \left(x\right) = 5 {x}^{4} - 6 {x}^{2} - 2 x$

 f''(x) = 20 x^3-12 x -2 ; f"(x)=0 , critical points are

$x = - 0.672 , x = - 0.176 , x = 0.847$ Let’s select a convenient

number in the interval less than $- 0.672$,

between −0.672 and -0.176, between −0.176 and 0.847

and greater than $0.847$. Let those be  −1, -0.5, 0.5 ,1

respectively . ${f}^{'} ' \left(- 1\right) \approx \left(-\right) , {f}^{'} ' \left(- 0.5\right) \approx \left(+\right)$ and

${f}^{'} ' \left(0.5\right) \approx \left(-\right) , {f}^{'} ' \left(1\right) \approx \left(+\right)$ , therefore , concave down

at $< - 0.672$ , concave up between −0.672 and -0.176,

concave down between $- 0.176 \mathmr{and} 0.847$ and

concave up at $\left(- \infty , - 0.672\right)$. Hence the The function has three

Inflection points at $x = - 0.672 , x = - 0.176 , x = 0.847$

graph{x^5-2x^3-x^2-2 [-20, 20, -10, 10]}

[Ans]