If there is a maximum and a minimum on a curve, will the inflection point between them be exactly in the middle? Or can it be located anywhere within the distance between the maximum and the minimum?
I don't think so.
Let
Then giving maxs/mins at x=0,-2b/(3a)
And giving infl pts at x=-b/(3a)
which is half way inbetween.
Now even if we change that to
the inflection pt is the same (x that is)
while the max/min is the solution to
which gives and that average is once again -b/(3a)
So it is true for polynomials of degree 3.
Moving onto the next power...
Try giving mins at x=0 and 2 and a max at 1
while giving infl pts at
and those two inflection pts are not at .5 and 1.5, the midpoints of 0 and 1, and of 1 and 2.