**Step 1:** Recall the relationship between the derivative of a function and its increasing/decreasing behavior.
A function is strictly increasing on an interval if its derivative is strictly positive on that interval.
**Step 2:** Analyze the given options in light of this relationship.
Option A states \(f^{\prime}(x) \lt 0\), which implies the function is strictly decreasing.
Option B states \(f^{\prime}(x) \gt 0\), which implies the function is strictly increasing.
Option C states \(f^{\prime}(x) = 0\), which implies the function is constant.
Option D is about the function's value, not its rate of change.
**Step 3:** Conclude based on the definition of a strictly increasing function.
For \(f(x)\) to be strictly increasing in \((a, b)\), its derivative \(f^{\prime}(x)\) must be greater than 0 for all \(x\) in \((a, b)\).
The final answer is $\boxed{B}$.
AI Tutor Explanation
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Step-by-Step Solution
Recall the condition for a strictly increasing function: A function \(f(x)\) is strictly increasing on an interval (a, b) if its derivative \(f^{\prime}(x)\) is positive for all x in that interval.
Evaluate the options:
(A) \(f^{\prime}(x) \lt 0\) implies the function is strictly decreasing.
(B) \(f^{\prime}(x) \gt 0\) implies the function is strictly increasing.
(C) \(f^{\prime}(x) = 0\) implies the function is constant.
(D) \(f(x) \gt 0\) does not provide information about whether the function is increasing or decreasing; it only indicates that the function's values are positive.
Select the correct option: Based on the condition for a strictly increasing function, option (B) is the correct choice.
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Pedagogical Audit
Bloom's Analysis:
This is an UNDERSTAND question because it requires the student to recall and explain the condition for a function to be strictly increasing based on its derivative.
Knowledge Dimension:CONCEPTUAL
Justification:The question tests the understanding of the concept of strictly increasing functions and their relationship with the sign of the first derivative.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's knowledge of the conditions for increasing/decreasing functions, a standard topic in the application of derivatives.