Let $f(x)$ be a real-valued function continuous on $[a, b]$ and differentiable on $(a, b)$. Which of the following statements is/are TRUE according to classical differential calculus theorems?
Options:
(A) If $f(a) = f(b)$, there exists at least one $c \in (a, b)$ such that $f'(c) = 0$ (Rolle's Theorem)
(B) There exists at least one $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$ (Lagrange's Mean Value Theorem)
(C) If $f'(x) = 0$ for all $x \in (a, b)$, then $f(x)$ is a constant function on $[a, b]$
(D) If $f'(x) > 0$ for all $x \in (a, b)$, then $f(x)$ is strictly increasing on $[a, b]$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
All four statements represent fundamental theorems of single-variable differential calculus:
1. Statement A is the exact statement of Rolle's theorem.
2. Statement B is Lagrange's Mean Value Theorem (LMVT).
3. Statement C is a direct corollary of LMVT: $f(x) - f(a) = f'(c)(x - a) = 0 \implies f(x) = f(a)$.
4. Statement D follows from LMVT: for $x_1 < x_2$, $f(x_2) - f(x_1) = f'(c)(x_2 - x_1) > 0 \implies f(x_2) > f(x_1)$.
Hence, A, B, C, and D are all correct.