Question

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

Answer 1

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(x) = x^5 - 7x^3-x^2-2`

`f'(x)=5x^4-21x^2-2x`

`f''(x)=20x^3-42x-2`

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

`20x^3-42x-2=0`

`10x^3-21x-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!