Why do the solutions to the derivatives of polynomials have the same x value as their peaks and valleys?

Question

I was graphing polynomials and their derivatives, and I noticed that the local maximums and minimums of of polynomials have the same x value as it's derivative's solutions. Is this just a coincidence?

For example, the polynomial, $3x^{5}+x^{4}+0.4x^{3}+x^{2}+2$, has a peak at, $-0.555$. The derivative of that polynomial is, $15x^4+4x^3+1.2x^2+2x$. It has the solution, $x = -0.555$

Answer 1

Note: The question has been changed. The following answers the new question: "Why do local extremas of a differentiable function $f(x)$ occur precisely where $f'(x)=0$?"

Suppose we have a differentiable function $f(x)$, and further suppose that $f(x)$ has a local max (or local min) at $x=c$. Then it must be that the slope of the tangent line to $f(x)$ at $x=c$ is $0$; that is, $f(x)$ has a horizontal tangent at $x=c$.

The derivative $f'(x)$ tells us the slope of the tangent line to $f(x)$ at any $x$-value we want. Therefore, it must be that $f'(c)=0$, since the we know that $f(x)$ has a tangent slope of $0$ at $x=c$.

In summary, if $f(x)$ has a local extrema at $x=c$, then $f'(c)=0$.

(However, if $f'(c)=0$, this does not necessarily mean that $f(x)$ has a local extrema at $x=c$; for example, $f(x)=x^3$ at $x=0$)

Answer 2

It is partly coincidence, partly not.

Take a polynomial that has at least three roots, and look at three consecutive roots of that polynomial. Like your example $x^6+x^5+x^4+2x^3+x^2$ and its roots $-1, -0.6832,0$. Between the first two roots $-1$ and $-0.6832$ there must be a critical point (in this case a local minimum), and between the last two roots $-0.6832$ and $0$ there must be a critical point (in this case a local maximum).

These two critical points correspond to roots of the derivative. And between the two roots of the derivative there must be a critical point of the derivative.

So somewhere between the two critical points of the original polynomial are both a root of the polynomial and a critical point of the derivative. There is no reason these two things need to coincide (although it's entirely possible to find examples where they do), but they might often be close enough together that it's difficult to tell from visual inspection of the graph that they are different. In your example, the critical point of the derivative actually happens at $x\approx -0.7103$, not $-0.6832$.