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 $\displaystyle y=ax^3+bx^2$
Then $\displaystyle y'=3ax^2+2bx$ giving maxs/mins at x=0,-2b/(3a)
And $\displaystyle y''=6ax+2b$ giving infl pts at x=-b/(3a)
which is half way inbetween.
Now even if we change that to $\displaystyle y=ax^3+bx^2+cx+d$
the inflection pt is the same (x that is)
while the max/min is the solution to $\displaystyle 3ax^2+2bx+c=0$
which gives $\displaystyle x={-2b\pm\sqrt{4b^2-12ac}\over 6a}$ and that average is once again -b/(3a)
So it is true for polynomials of degree 3.
Moving onto the next power...
Try $\displaystyle y'=x^3-3x^2+2x$ giving mins at x=0 and 2 and a max at 1
while $\displaystyle y''=3x^2-6x+2$ giving infl pts at $\displaystyle x=1\pm 1/\sqrt{3}$
and those two inflection pts are not at .5 and 1.5, the midpoints of 0 and 1, and of 1 and 2.