Class CBSE Class 12 Mathematics Applications of Derivatives Q #616
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
If \(f(x)=2x+\cos x\), then f(x):
(A) has a maxima at \(x=\pi\)
(B) has a minima at \(x=\pi\)
(C) is an increasing function
(D) is a decreasing function
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Correct Answer: C
Explanation
We differentiate $f(x)$ with respect to $x$:$$f'(x) = 2 - \sin x$$
We know that the range of the sine function is $-1 \le \sin x \le 1$.
This means that $f'(x)$ is always positive ($f'(x) > 0$) for all real values of $x$.

Since the first derivative $f'(x)$ is strictly positive for all $x$, the function $f(x) = 2x + \cos x$ is an increasing function throughout its domain.
This also rules out (A) and (B) because an increasing function has no local maxima or minima.

AI Tutor Explanation

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Step-by-Step Solution

  1. First, find the derivative of the function \(f(x) = 2x + \cos x\).

    \(f'(x) = \frac{d}{dx}(2x + \cos x) = 2 - \sin x\)

  2. Next, analyze the sign of \(f'(x)\). Since \(-1 \leq \sin x \leq 1\), we have:

    \(-1 \leq -\sin x \leq 1\)

    \(2 - 1 \leq 2 - \sin x \leq 2 + 1\)

    \(1 \leq f'(x) \leq 3\)

  3. Since \(f'(x) \geq 1\) for all \(x\), \(f'(x)\) is always positive. This means that the function \(f(x)\) is always increasing.

  4. Therefore, \(f(x)\) is an increasing function.

Correct Answer: is an increasing function

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concept of derivatives to determine whether the function is increasing or decreasing.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a series of steps, including finding the derivative and analyzing its sign, to determine the nature of the function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of how to use derivatives to analyze the increasing or decreasing nature of a function, a concept covered in the textbook.

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