gen-state-the-mean-value-theorem-precisely-and-explain-why-it-ma-e29d

State the mean value theorem precisely and explain why it matters in 2-3 sentences.

## Mean Value Theorem **Precise Statement:** If *f* is continuous on the closed interval [*a*, *b*] and differentiable on the open interval (*a*, *b*), then there exists at least one point *c* in (*a*, *b*) such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$ **Why it matters:** The MVT guarantees that somewhere between any two points on a smooth curve, the instantaneous rate of change (derivative) equals the average rate of change. This fundamental result bridges local and global behavior of functions, providing the theoretical foundation for many calculus theorems and practical applications like proving that if a derivative is zero everywhere, the function must be constant.