Class CBSE Class 12 Mathematics Applications of Derivatives Q #619
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The function \(f(x)=x^{2}-4x+6\) is increasing in the interval
(A) \((0, 2)\)
(B) \((-\infty, 2]\)
(C) \([1, 2]\)
(D) \([2, \infty)\)
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Correct Answer: D
Explanation
\[f(x) = x^2 - 4x + 6\]
\[f'(x) = 2x - 4\]
For Critical Point set \(f'(x) = 0\) implies \[2x - 4 = 0 \implies 2x = 4 \implies x = 2\]

The function is increasing when \(f'(x) > 0\) implies \[2x - 4 > 0\]\[2x > 4\]\[\mathbf{x > 2}\]

AI Tutor Explanation

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Step-by-Step Solution

  1. Find the derivative of the function \(f(x) = x^2 - 4x + 6\).

    \(f'(x) = 2x - 4\)

  2. To find the interval where the function is increasing, set \(f'(x) > 0\).

    \(2x - 4 > 0\)

  3. Solve the inequality for \(x\).

    \(2x > 4\)

    \(x > 2\)

  4. The function is increasing for \(x > 2\), which corresponds to the interval \((2, \infty)\). Since the options include closed intervals, we consider \([2, \infty)\) as the correct interval.

Correct Answer: [2, ∞)

AI Suggestion: Option D

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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 the intervals where the function is increasing. They must calculate the derivative, set it greater than zero, and solve for x.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific procedure: finding the derivative, setting it greater than zero, and solving the inequality to find the interval where the function is increasing.
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 related to increasing and decreasing functions, which is a standard topic in the syllabus.

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