Question

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 ?

Answer 1

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 `d/(dx)p_(21)(x,-2)`. As we can observe `d/(dx)p_(21)(x,-2) = 0` does not have real roots, then `p_(21)(x,-2)` has not relative maxima/minima.

And finally the plot for `d^2/(dx^2)p_(21)(x,-2)` showing the inflexion points location. as the roots of `d^2/(dx^2)p_(21)(x,-2) = 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.