Part 4/7:
Rolle's Theorem is particularly relevant in identifying local maximum and minimum values of continuous and differentiable functions. A local maximum occurs at points where the slope of the function transitions from positive to negative, while local minima occur where it shifts from negative to positive. In both cases, this change is accompanied by ( f'(c) = 0 ).
An illustrative example involves the motion of an object, such as a ball thrown into the air. The maximum height corresponds to the point where the ball's velocity drops to zero, reaffirming the relationship between velocity, position, and Rolle's Theorem.