For a polynomial function the instantaneous rate of change could be used to locate local maximum and local minimum because when the instantaneous rate of change is 0 there is a local maximum or minimum.
Or in other words when the instantaneous rate of change goes from negative to positive, there are a local minimum of the function. (when the rate of change is 0)
And when the instantaneous rate of change goes from positive to negative, there are a local maximum of the function.
Remember that the instantaneous rate of change represents the slope of the tangent line to the cuve at that point. Slope zero represents a horizontal tangent line and that only happens when there is a local minimum or maximum.