When the Speedometer Matches the Trip

A quantity follows the curve

$f(x) = x^3 - 6x$

Across the interval $[0,\,3]$ this curve has one average rate of change — the "whole trip" number. At each individual point it has an instantaneous rate of change — the "speedometer" number, obtained by letting the gap $h$ in

$\frac{f(a+h) - f(a)}{h}$

shrink toward $0$.

Somewhere strictly inside the interval there is a point $c$ where the speedometer reads exactly the trip average. Find that $c$ in $(0,\,3)$, and state the common rate.