Class CBSE Class 12 Mathematics Applications of Derivatives Q #1380
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2025 AISSCE(Board Exam) VSA
Find the values of 'a' for which $f(x)=\sin x-ax+b$ is increasing on R.
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Detailed Solution

Step 1: Understand the condition for increasing function

A function $f(x)$ is increasing on R if its derivative $f'(x) \ge 0$ for all $x \in R$.

Step 2: Find the derivative of f(x)

Given $f(x) = \sin x - ax + b$, we need to find its derivative $f'(x)$. $$f'(x) = \frac{d}{dx}(\sin x - ax + b) = \cos x - a$$

Step 3: Apply the increasing function condition

For $f(x)$ to be increasing on R, we must have $f'(x) \ge 0$ for all $x \in R$. $$\cos x - a \ge 0$$ $$a \le \cos x$$

Step 4: Determine the range of cos x

We know that the range of $\cos x$ is $[-1, 1]$. Therefore, the maximum value of $\cos x$ is 1 and the minimum value is -1. $$-1 \le \cos x \le 1$$

Step 5: Find the condition on 'a'

Since $a \le \cos x$ for all $x \in R$, $a$ must be less than or equal to the minimum value of $\cos x$. However, we need $a \le \cos x$ to hold true for all $x$. Therefore, $a$ must be less than or equal to the *minimum* value of $\cos x$, which is -1. $$a \le -1$$

Final Answer: $a \le -1$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of increasing functions and how it relates to the derivative, and then apply this understanding to the given function.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the conceptual relationship between the derivative of a function and its increasing/decreasing behavior. It involves applying the concept that $f'(x) \ge 0$ for an increasing function.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the application of derivatives to determine the intervals where a function is increasing or decreasing, a standard topic in the application of derivatives.

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