Rolle’s Theorem

Rolle’s Theorem states that if a function is continuous on a closed interval, differentiable on the open interval, and equal at both endpoints, then the derivative is zero at some point between the endpoints.

Consider a road over which a vehicle climbs up, reaches a peak, and then descends.

Since it starts and ends at the same height, there must be a point where the ascent changes to a descent. At that point, the slope becomes zero, satisfying Rolle’s theorem.

A function on a closed interval can take various shapes, all of which may satisfy Rolle’s Theorem if the conditions are met.

Some functions may have more than one point where the derivative is zero, like when there are both local maxima and minima within the interval.

On the other hand, the altitude of a train on a flat track is represented graphically as a horizontal line. Here, every point along the track satisfies Rolle’s Theorem, as the derivative is zero everywhere on this line.