JEE Exam > JEE Questions > Let f (x) =x3−6x2+9x+8, then f (x) is d...
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Let f (x) = x3−6x2+9x+8, then f (x) is decreasing in
- a)(−∞,1) ∪ (3,∞)
- b)[1, 3]
- c)[3,∞]
- d)(−∞,1)
Correct answer is option 'B'. Can you explain this answer?
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Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−...
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Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−...
Understanding the Function f(x)
The function given is f(x) = x^3 - 6x^2 + 9x + 8. To determine where this function is decreasing, we need to analyze its first derivative.
Finding the First Derivative
- The first derivative f'(x) helps identify increasing and decreasing intervals.
- f'(x) = 3x^2 - 12x + 9.
Finding Critical Points
- Set the first derivative equal to zero to find critical points:
3x^2 - 12x + 9 = 0.
- Simplifying gives us:
x^2 - 4x + 3 = 0.
- Factoring results in:
(x - 1)(x - 3) = 0, leading to critical points at x = 1 and x = 3.
Test Intervals
Now we need to evaluate the sign of f'(x) in the intervals defined by these critical points:
- Interval (-∞, 1): Choose x = 0.
f'(0) = 3(0)^2 - 12(0) + 9 = 9 (positive).
- Interval (1, 3): Choose x = 2.
f'(2) = 3(2)^2 - 12(2) + 9 = -3 (negative).
- Interval (3, ∞): Choose x = 4.
f'(4) = 3(4)^2 - 12(4) + 9 = 9 (positive).
Conclusion
- f'(x) > 0 in (-∞, 1) and (3, ∞) (increasing).
- f'(x) < 0="" in="" (1,="" 3)="" />
Thus, f(x) is decreasing in the interval [1, 3], which corresponds to option 'B'. 0="" in="" (1,="" 3)="" (decreasing).="" thus,="" f(x)="" is="" decreasing="" in="" the="" interval="" [1,="" 3],="" which="" corresponds="" to="" option="">
Thus, f(x) is decreasing in the interval [1, 3], which corresponds to option 'B'.>
The function given is f(x) = x^3 - 6x^2 + 9x + 8. To determine where this function is decreasing, we need to analyze its first derivative.
Finding the First Derivative
- The first derivative f'(x) helps identify increasing and decreasing intervals.
- f'(x) = 3x^2 - 12x + 9.
Finding Critical Points
- Set the first derivative equal to zero to find critical points:
3x^2 - 12x + 9 = 0.
- Simplifying gives us:
x^2 - 4x + 3 = 0.
- Factoring results in:
(x - 1)(x - 3) = 0, leading to critical points at x = 1 and x = 3.
Test Intervals
Now we need to evaluate the sign of f'(x) in the intervals defined by these critical points:
- Interval (-∞, 1): Choose x = 0.
f'(0) = 3(0)^2 - 12(0) + 9 = 9 (positive).
- Interval (1, 3): Choose x = 2.
f'(2) = 3(2)^2 - 12(2) + 9 = -3 (negative).
- Interval (3, ∞): Choose x = 4.
f'(4) = 3(4)^2 - 12(4) + 9 = 9 (positive).
Conclusion
- f'(x) > 0 in (-∞, 1) and (3, ∞) (increasing).
- f'(x) < 0="" in="" (1,="" 3)="" />
Thus, f(x) is decreasing in the interval [1, 3], which corresponds to option 'B'. 0="" in="" (1,="" 3)="" (decreasing).="" thus,="" f(x)="" is="" decreasing="" in="" the="" interval="" [1,="" 3],="" which="" corresponds="" to="" option="">
Thus, f(x) is decreasing in the interval [1, 3], which corresponds to option 'B'.>
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Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−∞,1) ∪ (3,∞)b)[1, 3]c)[3,∞]d)(−∞,1)Correct answer is option 'B'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−∞,1) ∪ (3,∞)b)[1, 3]c)[3,∞]d)(−∞,1)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−∞,1) ∪ (3,∞)b)[1, 3]c)[3,∞]d)(−∞,1)Correct answer is option 'B'. Can you explain this answer?.
Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−∞,1) ∪ (3,∞)b)[1, 3]c)[3,∞]d)(−∞,1)Correct answer is option 'B'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−∞,1) ∪ (3,∞)b)[1, 3]c)[3,∞]d)(−∞,1)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f (x) =x3−6x2+9x+8, then f (x) is decreasing ina)(−∞,1) ∪ (3,∞)b)[1, 3]c)[3,∞]d)(−∞,1)Correct answer is option 'B'. Can you explain this answer?.
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