What are the points of inflection, if any, of #f(x)= x^5 -7 x^3- x^2-2 #?
We can't easily find exact locations, but there are three inflection points at roughly x = -1.4, 0, and 1.5. Graphs below.
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.
Set f''(x) = 0 (since for polynomials the derivatives are never undefined):
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!