Rolle's Theorem

Rolle's Theorem states that if a smooth, continuous curve (representing a function) starts and ends at the same value over a specific interval, then there must be at least one point in between where the slope of the curve is zero—that is, a flat spot. Essentially, if a graph goes up and comes back down to the same height without any jumps or breaks, it must have at least one peak or valley where the tangent line is horizontal. This theorem helps us understand how functions behave and guarantees the existence of stationary points under certain conditions.