Question

Let \( f(x) \) be a continuous function on \([a, b]\) and differentiable on \((a, b)\). Then, this function \( f(x) \) is strictly increasing in \((a, b)\) if:

• A. \( f'(x) < 0, \ \forall \ x \in (a, b) \)
• B. \( f'(x) > 0, \ \forall \ x \in (a, b) \)
• C. \( f'(x) = 0, \ \forall \ x \in (a, b) \)
• D. \( f(x) > 0, \ \forall \ x \in (a, b) \)
• E.

Hint:

A function is strictly increasing in an interval if its derivative is positive throughout the interval. Always check the sign of the derivative for such conditions.

Answer 1

The Correct Option is B

To establish the conditions for a function \( f(x) \) to be strictly increasing on the interval \((a, b)\), we use the following mathematical criterion:1. Definition of a Strictly Increasing Function: A function \( f(x) \) is defined as strictly increasing on an interval \((a, b)\) if its derivative satisfies: \[ f'(x) > 0, \ \forall \ x \in (a, b). \] This signifies that the derivative of \( f(x) \) must maintain a positive value across the entire interval.2. Evaluation of Options: - Option (A): \( f'(x) < 0, \ \forall \ x \in (a, b) \) indicates that \( f(x) \) is strictly decreasing, thus it is incorrect. - Option (B): \( f'(x) > 0, \ \forall \ x \in (a, b) \) accurately implies that \( f(x) \) is strictly increasing, thus it is correct. - Option (C): \( f'(x) = 0, \ \forall \ x \in (a, b) \) implies that \( f(x) \) is constant, not strictly increasing, thus it is incorrect. - Option (D): \( f(x) > 0, \ \forall \ x \in (a, b) \) does not confirm that \( f(x) \) is strictly increasing, as the condition \( f(x) > 0 \) does not describe the derivative's behavior, thus it is incorrect.3. Conclusion: The requisite condition for \( f(x) \) to be strictly increasing on \((a, b)\) is that the derivative \( f'(x) \) must fulfill: \[ f'(x) > 0, \ \forall \ x \in (a, b). \] Therefore, the correct selection is (B) \( f'(x) > 0, \ \forall \ x \in (a, b) \).