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 determine the conditions under which a function \( f(x) \) is strictly increasing on an interval \((a, b)\), we use the following mathematical definition: 1. Strictly Increasing Function: - A function \( f(x) \) is said to be strictly increasing on an interval \((a, b)\) if: \[ f'(x) > 0, \ \forall \ x \in (a, b). \] - This means that the derivative of \( f(x) \) must be positive throughout the interval. 2. Analysis of Options: - (A): \( f'(x) < 0, \ \forall \ x \in (a, b) \) implies \( f(x) \) is strictly decreasing, not increasing. Hence, incorrect. - (B): \( f'(x) > 0, \ \forall \ x \in (a, b) \) correctly implies that \( f(x) \) is strictly increasing. Hence, correct. - (C): \( f'(x) = 0, \ \forall \ x \in (a, b) \) implies that \( f(x) \) is constant, not strictly increasing. Hence, incorrect. - (D): \( f(x) > 0, \ \forall \ x \in (a, b) \) does not guarantee that \( f(x) \) is strictly increasing because \( f(x) > 0 \) does not describe the behavior of the derivative. Hence, incorrect. 3. Conclusion: For \( f(x) \) to be strictly increasing on \((a, b)\), the derivative \( f'(x) \) must satisfy: \[ f'(x) > 0, \ \forall \ x \in (a, b). \] Hence, the correct answer is (B) \( f'(x) > 0, \ \forall \ x \in (a, b) \).