# What are the points of inflection, if any, of f(x)= x^5 -7 x^3- x^2-2 ?

Feb 6, 2016

We can't easily find exact locations, but there are three inflection points at roughly x = -1.4, 0, and 1.5. Graphs below.

#### Explanation:

To find inflection points we want the concavity to change from concave up to concave down, usually the second derivative is zero (or undefined) there.

$f \left(x\right) = {x}^{5} - 7 {x}^{3} - {x}^{2} - 2$

$f ' \left(x\right) = 5 {x}^{4} - 21 {x}^{2} - 2 x$

$f ' ' \left(x\right) = 20 {x}^{3} - 42 x - 2$

Set f''(x) = 0 (since for polynomials the derivatives are never undefined):

$20 {x}^{3} - 42 x - 2 = 0$

$10 {x}^{3} - 21 x - 1 = 0$

This f''(x) = 0 equation doesn't factor or have rational roots, but does have three solutions, at about -1.42, -0.04, and 1.47. (I used tables of values to zoom in on the roots.)

The original function with its three inflection points:

// dansmath \\ strikes again!