Q1401: If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that f is an increasi...

CBSE Class 12 Mathematics Applications of Derivatives Q #1401

KNOWLEDGE BASED

UNDERSTAND

2 Marks 2025 AISSCE(Board Exam) VSA

If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that f is an increasing function.

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AI Tutor Explanation

Detailed Solution

Step 1: Find the derivative of f(x)

Given $f(x) = x + \frac{1}{x}$, we need to find its derivative $f'(x)$. Differentiating with respect to $x$, we get: $$f'(x) = \frac{d}{dx}(x + \frac{1}{x}) = \frac{d}{dx}(x) + \frac{d}{dx}(x^{-1})$$ $$f'(x) = 1 + (-1)x^{-2} = 1 - \frac{1}{x^2}$$

Step 2: Analyze the sign of f'(x) for x ≥ 1

We need to show that $f'(x) > 0$ for $x \ge 1$. $$f'(x) = 1 - \frac{1}{x^2}$$ Since $x \ge 1$, we have $x^2 \ge 1$. Therefore, $\frac{1}{x^2} \le 1$. $$1 - \frac{1}{x^2} \ge 0$$ Thus, $f'(x) \ge 0$ for $x \ge 1$.

Step 3: Conclude that f(x) is increasing

Since $f'(x) \ge 0$ for $x \ge 1$, the function $f(x)$ is an increasing function for $x \ge 1$.

Final Answer: f(x) is an increasing function for x ≥ 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 to determine if a function is increasing using its derivative.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concept of derivatives and their relationship to increasing functions, rather than just recalling facts or performing a routine procedure.<\/span>

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly relates to the textbook content on application of derivatives to determine increasing/decreasing functions.