# Are there polynomial functions whose graphs have: 11 points of inflection, but no max or min ?

## Are there polynomial functions whose graphs have: 1001 points of inflection, but no max or min ?

Feb 12, 2018

See below.

#### Explanation:

You can build a polynomial with as many inflexion points as needed using the truncated series expansion for sinx = sum_(k=0)^n (-1)^kx^(2k-1)/((2k-1)!) added to a line with convenient gradient or as

p_n(x,m) = sum_(k=0)^n (-1)^kx^(2k-1)/((2k-1)!)+ m x

For instance, an example for $m = - 2$ and $n = 21$ with exactly $11$ inflexion points.

The next plot shows $\frac{d}{\mathrm{dx}} {p}_{21} \left(x , - 2\right)$. As we can observe $\frac{d}{\mathrm{dx}} {p}_{21} \left(x , - 2\right) = 0$ does not have real roots, then ${p}_{21} \left(x , - 2\right)$ has not relative maxima/minima.

And finally the plot for ${d}^{2} / \left({\mathrm{dx}}^{2}\right) {p}_{21} \left(x , - 2\right)$ showing the inflexion points location. as the roots of ${d}^{2} / \left({\mathrm{dx}}^{2}\right) {p}_{21} \left(x , - 2\right) = 0$ . Here we can count exactly $11$ inflexion points,

NOTE

Depending on $n$ the sign for $m$ can be positive or negative, and $n > 4$. It is left as an exercise to determine the connection for the $m$ sign with the $n$ value as well as the dependency between the sought inflection points number, with the $n$ value.