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 ?

1 Answer
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.

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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.

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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,

enter image source here

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.