Class CBSE Class 12 Mathematics Applications of Derivatives Q #932
COMPETENCY BASED
APPLY
2 Marks 2023 VSA
If \(f(x)=a(\tan x-\cot x)\), where \(a>0\), then find whether \(f(x)\) is increasing or decreasing function in its domain.
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Step-by-Step Solution

  1. Step 1: Differentiate the function \(f(x)\) with respect to \(x\).

    Given \(f(x) = a(\tan x - \cot x)\), we need to find \(f'(x)\).

    \(f'(x) = a(\sec^2 x - (-\csc^2 x)) = a(\sec^2 x + \csc^2 x)\)

  2. Step 2: Analyze the sign of \(f'(x)\).

    Since \(a > 0\), \(\sec^2 x\) is always positive, and \(\csc^2 x\) is always positive, their sum is also always positive.

    Therefore, \(f'(x) = a(\sec^2 x + \csc^2 x) > 0\) for all \(x\) in the domain of \(f(x)\).

  3. Step 3: Conclude whether \(f(x)\) is increasing or decreasing.

    Since \(f'(x) > 0\) for all \(x\) in the domain, the function \(f(x)\) is an increasing function.

Correct Answer: Increasing Function<\/strong>

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of differentiation and increasing/decreasing functions to determine the nature of the given function.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure (differentiation, analysis of the derivative's sign) to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply calculus concepts to analyze the behavior of a function, which aligns with competency-based assessment.

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