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The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in [0, 2] is:
APPLY KNOWLEDGE 1 Marks
Concept Application
50%
Calculation / Logic
50%
Target Level
MEDIUM
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APPLY KNOWLEDGE MEDIUM

Q: The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in [0, 2] is:

Question Analysis & Solution

Step-by-Step Solution

To find the absolute maximum value of the function \( f(x) = x^3 - 3x + 2 \) in the interval [0, 2], we need to follow these steps: 1. **Find the critical points:** * Take the derivative of \( f(x) \): \[ f'(x) = 3x^2 - 3 \] * Set \( f'(x) = 0 \) and solve for \( x \): \[ 3x^2 - 3 = 0 \] \[ 3x^2 = 3 \] \[ x^2 = 1 \] \[ x = \pm 1 \] * The critical points are \( x = 1 \) and \( x = -1 \). Since we are only considering the interval [0, 2], we only consider \( x = 1 \). 2. **Evaluate the function at the critical points and endpoints:** * Evaluate \( f(x) \) at \( x = 0 \), \( x = 1 \), and \( x = 2 \): * \( f(0) = (0)^3 - 3(0) + 2 = 2 \) * \( f(1) = (1)^3 - 3(1) + 2 = 1 - 3 + 2 = 0 \) * \( f(2) = (2)^3 - 3(2) + 2 = 8 - 6 + 2 = 4 \) 3. **Determine the absolute maximum value:** * Comparing the values \( f(0) = 2 \), \( f(1) = 0 \), and \( f(2) = 4 \), the absolute maximum value is 4.

Correct Answer: 4

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply the concepts of derivatives and finding critical points to determine the absolute maximum value of a function within a given interval.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a step-by-step procedure to find the derivative, critical points, and evaluate the function at those points and endpoints.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of calculus concepts taught in the textbook.
AI Suggestion: Option C
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