Q1254: Find the absolute maximum and absolute minimum values of the...

Class CBSE Class 12 Mathematics Applications of Derivatives Q #1254

KNOWLEDGE BASED

REMEMBER

3 Marks 2024 AISSCE(Board Exam) SA

Find the absolute maximum and absolute minimum values of the function f given by $f(x)=\frac{x}{2}+\frac{2}{x}$ , on the interval [1, 2].

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AI Tutor Explanation

Detailed Solution

Step 1: Find the derivative of f(x)

We are given the function $f(x) = \frac{x}{2} + \frac{2}{x}$. To find the absolute maximum and minimum values, we first need to find the critical points by taking the derivative and setting it to zero.

Step 2: Calculate f'(x)

The derivative of $f(x)$ is: $$f'(x) = \frac{d}{dx} \left(\frac{x}{2} + \frac{2}{x}\right) = \frac{1}{2} - \frac{2}{x^2}$$

Step 3: Find critical points

Set $f'(x) = 0$ to find the critical points: $$\frac{1}{2} - \frac{2}{x^2} = 0$$ $$\frac{1}{2} = \frac{2}{x^2}$$ $$x^2 = 4$$ $$x = \pm 2$$

Step 4: Check critical points within the interval [1, 2]

Since we are considering the interval [1, 2], we only consider the critical point $x = 2$ because $x = -2$ is not in the interval.

Step 5: Evaluate f(x) at the endpoints and critical points

Now, we evaluate $f(x)$ at the endpoints of the interval (1 and 2) and at the critical point $x = 2$: $$f(1) = \frac{1}{2} + \frac{2}{1} = \frac{1}{2} + 2 = \frac{5}{2} = 2.5$$ $$f(2) = \frac{2}{2} + \frac{2}{2} = 1 + 1 = 2$$

Step 6: Determine absolute maximum and minimum values

Comparing the values, we find that the absolute maximum value is $f(1) = \frac{5}{2}$ and the absolute minimum value is $f(2) = 2$.

Final Answer: Absolute maximum value: 5/2, Absolute minimum value: 2

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Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because the student needs to recall the procedure for finding absolute maximum and minimum values of a function on a closed interval. This involves finding the derivative, critical points, and evaluating the function at the endpoints and critical points.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific algorithm or method (finding derivatives, critical points, and evaluating the function) to solve the problem.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the concepts and procedures related to finding absolute maximum and minimum values, which is a standard topic in the application of derivatives.