Q932: If $f(x)=a(\tan x-\cot x)$, where $a>0$, then find wheth...

Class CBSE Class 12 Mathematics Applications of Derivatives Q #932

KNOWLEDGE BASED

UNDERSTAND

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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AI Tutor Explanation

Detailed Solution

Step 1: Rewrite the function

We are given the function $f(x) = a(\tan x - \cot x)$, where $a > 0$. We can rewrite $\cot x$ as $\frac{1}{\tan x}$, so the function becomes: $$f(x) = a\left(\tan x - \frac{1}{\tan x}\right)$$

Step 2: Differentiate the function

To determine whether the function is increasing or decreasing, we need to find its derivative, $f'(x)$. $$f'(x) = a\left(\frac{d}{dx}(\tan x) - \frac{d}{dx}(\cot x)\right)$$ We know that $\frac{d}{dx}(\tan x) = \sec^2 x$ and $\frac{d}{dx}(\cot x) = -\csc^2 x$. Therefore, $$f'(x) = a(\sec^2 x - (-\csc^2 x)) = a(\sec^2 x + \csc^2 x)$$

Step 3: Analyze the sign of the derivative

Since $a > 0$, we need to determine the sign of $\sec^2 x + \csc^2 x$. We know that $\sec^2 x$ and $\csc^2 x$ are always non-negative (since they are squares). Also, they cannot both be zero at the same time. Therefore, $\sec^2 x + \csc^2 x > 0$ for all $x$ in the domain of $f(x)$. Since $a > 0$ and $\sec^2 x + \csc^2 x > 0$, we have $f'(x) = a(\sec^2 x + \csc^2 x) > 0$.

Step 4: Conclude whether the function is increasing or decreasing

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

Final Answer: Increasing function

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concepts of derivatives and increasing/decreasing functions to solve the problem. They must also understand trigonometric functions and their derivatives.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concepts of derivatives, trigonometric functions, and increasing/decreasing functions, rather than just recalling facts or applying a specific algorithm.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of derivatives and their application in determining the nature of functions (increasing or decreasing), which is a core concept in the syllabus.

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