Question

Let \( \mathcal{F} = \{ f: [a, b] \to \mathbb{R} \mid f \text{ is continuous on } [a, b] \text{ and differentiable on } (a, b) \} \).
Which one of the following options is correct?

• A. There exists a non-constant \( f \in \mathcal{F} \) such that \( |f(x) - f(y)| \leq |x - y|^2 \) for all \( x, y \in [a, b] \)
• B. If \( f \in \mathcal{F} \) and \( x_0 \in (a, b) \), then there exist distinct \( x_1, x_2 \in [a, b] \) such that \( \frac{f(x_1) - f(x_2)}{x_1 - x_2} = f'(x_0) \)
• C. Let \( f \in \mathcal{F} \) and \( f'(x) \geq 0 \) for all \( x \in (a, b) \). If \( f' \) is zero only at two distinct points, then \( f \) is strictly increasing.
• D. Let \( f \in \mathcal{F} \). If \( f'(x_1)<c<f'(x_2) \) for some \( x_1, x_2 \in (a, b) \), then there may NOT exist an \( x_0 \in (x_1, x_2) \) such that \( f'(x_0) = c \).

Hint:

The Mean Value Theorem guarantees the existence of a point where the average rate of change equals the instantaneous rate of change. Use this to answer questions involving the relationship between function values and derivatives.

Answer 1

The Correct Option is C

Step 1: Analyzing option (A).
The condition \( |f(x) - f(y)| \leq |x - y|^2 \) implies that \( f \) must be a function that grows slower than linearly with respect to the difference \( |x - y| \), i.e., \( f \) must be of the form \( f(x) = kx^2 \) for some constant \( k \), which is a valid option. However, it is not necessarily true for every non-constant function in \( \mathcal{F} \). Hence, option (A) is not universally valid.

Step 2: Analyzing option (B).
This option is correct. The Mean Value Theorem tells us that for a function \( f \) continuous on the closed interval \( [a, b] \) and differentiable on the open interval \( (a, b) \), there exists a point \( c \in (a, b) \) such that: \[ \frac{f(x_1) - f(x_2)}{x_1 - x_2} = f'(c). \] Thus, for \( x_1, x_2 \in [a, b] \) with \( x_1 \neq x_2 \), there exist points \( x_1 \) and \( x_2 \) where the Mean Value Theorem applies, ensuring that \( \frac{f(x_1) - f(x_2)}{x_1 - x_2} = f'(x_0) \).

Step 3: Analyzing option (C).
If \( f'(x) \geq 0 \) for all \( x \in (a, b) \), and \( f' \) is zero only at two distinct points, then \( f \) is non-decreasing, but it is not necessarily strictly increasing. For \( f \) to be strictly increasing, \( f'(x) \) must be strictly positive for all \( x \) except at isolated points. Hence, option (C) is incorrect as it overstates the conclusion.

Step 4: Analyzing option (D).
If \( f'(x_1) < c < f'(x_2) \), by the Intermediate Value Theorem, there must exist some \( x_0 \in (x_1, x_2) \) such that \( f'(x_0) = c \). Hence, option (D) is incorrect.

Thus, the correct answer is \( \boxed{(C)} \).