To find the interval where the function f(x) = 2x³ + 9x² + 12x - 1 is decreasing, we need to find where its derivative f'(x) is negative.
1. **Find the derivative f'(x):**
f'(x) = d/dx (2x³ + 9x² + 12x - 1)
f'(x) = 6x² + 18x + 12
2. **Find the critical points by setting f'(x) = 0:**
6x² + 18x + 12 = 0
Divide by 6:
x² + 3x + 2 = 0
Factor the quadratic equation:
(x + 1)(x + 2) = 0
So, x = -1 and x = -2 are the critical points.
3. **Determine the intervals where f'(x) is negative:**
We have three intervals to consider: (-∞, -2), (-2, -1), and (-1, ∞).
- For x < -2 (e.g., x = -3):
f'(-3) = 6(-3)² + 18(-3) + 12 = 6(9) - 54 + 12 = 54 - 54 + 12 = 12 > 0
So, f(x) is increasing in the interval (-∞, -2).
- For -2 < x < -1 (e.g., x = -1.5):
f'(-1.5) = 6(-1.5)² + 18(-1.5) + 12 = 6(2.25) - 27 + 12 = 13.5 - 27 + 12 = -1.5 < 0
So, f(x) is decreasing in the interval (-2, -1).
- For x > -1 (e.g., x = 0):
f'(0) = 6(0)² + 18(0) + 12 = 12 > 0
So, f(x) is increasing in the interval (-1, ∞).
4. **Identify the interval where f(x) is decreasing:**
The function f(x) is decreasing in the interval (-2, -1).
Correct Answer: (-2,-1)
Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires students to apply the concepts of derivatives and critical points to determine the interval of decreasing function.
Knowledge Dimension:PROCEDURAL
Justification:The question requires the student to apply a specific procedure (finding the derivative, setting it to zero, and testing intervals) 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 and application of calculus concepts taught in the textbook.
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