Determine where peak or valley of polynomial graph without calculus I notice that if I graph $y=(x+2)^2(x-4)^2$ that the midpoint of the roots occurs at $x=1$.  I note that this is also where the local maximum occurs.
If I graph $y=(x-5)^2(x^2)$ the midpoint of the roots is 2.5 where the local maximum occurs.
I also know how to find the exact values of the local max / min by taking the derivative and setting to zero.
My question is, for those in pre-calculus who have not yet learned about derivatives, is there a relationship between where the roots occur and the local max / min values?  Certainly for the 2 examples I chose, there seems to be a relationship, but I can't see the mathematical reasoning for this. Calculus explains this, but could I predict the local max/min by simply finding the mid-point of the roots?

There is no precise relation because you can keep the extrema fixed while the roots are moving. Consider the cubic
$$x^3-3x+c,$$ that always has a maximum at $x=-1$ and a minimum at $x=1$.
We can place a root wherever we want, say at $x_0$, just by setting
$$c=-x_0^3+3x_0.$$

