point of inflection -- always halfway between 2 critical points??

[brum]

when you look at the graph of a function
say, f(x) = 5x^3 - 2x^2 + 3x - 1

will the point(s) of inflection always be equidistant from 2 critical points (ie the 2 nearest critical points)???


point of inflection -- point where the concavity changes from up to down / down to up

critical point -- point where the function changes from increasing to decreasing or decreasing to increasing

[selfAdjoint]

Okay, this is about cubics. With any cubic you can do a linear transformation and produce a coordinate system in which the curve
a) passes through the origin and
b) is unicursal, it only passes through each coordinate line once, and
c) the part in the negative half of the plane is the reversed mirror image of the part in the positive half of the palne.

Now the cubic has at most two critical points and one inflection because its first derivative is a quadratic with two or no real roots, and its second derivative is linear with one root.

So take that one and only one inflection point. Where can it lie? Not in the negative half of the plane, because there's no second inflection point to match it on the positive side. And by the same reasoning not in the right half plane either. Therefore it lies on the y-axis, at x=0.

I think I'll leave the rest of the proof for you to finish.