Class CBSE Class 12 Mathematics Applications of Derivatives Q #611
KNOWLEDGE BASED
APPLY
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
The function \(f(x)=x^{3}-3x^{2}+12x-18\) is:
(A) strictly decreasing on R
(B) strictly increasing on R
(C) neither strictly increasing nor strictly decreasing on R
(D) strictly decreasing on \((-\infty, 0)\)
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Explanation
Here's the step-by-step solution:

1. **Find the first derivative of the function:**
$f'(x) = \frac{d}{dx}(x^3 - 3x^2 + 12x - 18) = 3x^2 - 6x + 12$.

2. **Analyze the sign of the first derivative:** The derivative is a quadratic function. To determine its sign, we can find its discriminant. The discriminant of $ax^2 + bx + c$ is $b^2 - 4ac$. For $3x^2 - 6x + 12$, the discriminant is $(-6)^2 - 4(3)(12) = 36 - 144 = -108$.

3. **Interpret the discriminant:** Since the discriminant is negative and the leading coefficient (3) is positive, the quadratic $f'(x) = 3x^2 - 6x + 12$ is always positive for all real values of $x$.

4. **Conclude the function's behavior:** Because $f'(x) > 0$ for all $x \in \mathbb{R}$, the function $f(x)$ is strictly increasing on $\mathbb{R}$.

The final answer is $\boxed{B}$.

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Step-by-Step Solution

  1. Step 1: Find the derivative of the function.

    Given \(f(x) = x^3 - 3x^2 + 12x - 18\), we find its derivative \(f'(x)\).

    \(f'(x) = 3x^2 - 6x + 12\)

  2. Step 2: Analyze the derivative.

    We want to determine if \(f'(x)\) is always positive, always negative, or changes sign.

    \(f'(x) = 3(x^2 - 2x + 4)\)

    Complete the square: \(f'(x) = 3((x - 1)^2 + 3)\)

  3. Step 3: Determine the sign of the derivative.

    Since \((x - 1)^2\) is always non-negative, \((x - 1)^2 + 3\) is always greater than or equal to 3. Therefore, \(3((x - 1)^2 + 3)\) is always positive.

    Thus, \(f'(x) > 0\) for all \(x \in \mathbb{R}\).

  4. Step 4: Conclude the nature of the function.

    Since \(f'(x) > 0\) for all \(x\), the function \(f(x)\) is strictly increasing on \(\mathbb{R}\).

Correct Answer: strictly increasing on R

AI Suggestion: Option B

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of derivatives to determine if a function is increasing or decreasing.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a series of steps (differentiation, finding critical points, and testing intervals) to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of how to use derivatives to analyze the increasing/decreasing nature of functions, a standard topic in the textbook.

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