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8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above?
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8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) c...
Function Definition
The function f(x) = |x| * |x + 1| * |x + 2| * |x - 1| * |x - 2| is composed of products of absolute values of linear functions.
Continuity
- The function f(x) is continuous for all x in R.
- Absolute value functions are continuous, and the product of continuous functions is also continuous.
Differentiability
- The function is not differentiable at points where any of the absolute value terms equal zero.
- The critical points where the absolute values change are at x = 0, -1, -2, 1, and 2.
Analysis at Critical Points
- At x = 0:
- The function transitions from negative to positive, indicating potential non-differentiability.
- At x = -1 and x = -2:
- Similar arguments apply; the absolute value terms change behavior.
- At x = 1 and x = 2:
- Again, the function's derivative may not exist due to changes in the slope.
Conclusion
- The function f(x) is:
- (a) Continuous in R: True
- (b) Differentiable in R: False
- (c) Only differentiable at x = 0, -1, -2: False
- (d) None of the above: True
Thus, the correct answer is (d) None of the above.
The function f(x) = |x| * |x + 1| * |x + 2| * |x - 1| * |x - 2| is composed of products of absolute values of linear functions.
Continuity
- The function f(x) is continuous for all x in R.
- Absolute value functions are continuous, and the product of continuous functions is also continuous.
Differentiability
- The function is not differentiable at points where any of the absolute value terms equal zero.
- The critical points where the absolute values change are at x = 0, -1, -2, 1, and 2.
Analysis at Critical Points
- At x = 0:
- The function transitions from negative to positive, indicating potential non-differentiability.
- At x = -1 and x = -2:
- Similar arguments apply; the absolute value terms change behavior.
- At x = 1 and x = 2:
- Again, the function's derivative may not exist due to changes in the slope.
Conclusion
- The function f(x) is:
- (a) Continuous in R: True
- (b) Differentiable in R: False
- (c) Only differentiable at x = 0, -1, -2: False
- (d) None of the above: True
Thus, the correct answer is (d) None of the above.
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8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above?
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8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above? for Teaching 2025 is part of Teaching preparation. The Question and answers have been prepared according to the Teaching exam syllabus. Information about 8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above? covers all topics & solutions for Teaching 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above?.
8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above? for Teaching 2025 is part of Teaching preparation. The Question and answers have been prepared according to the Teaching exam syllabus. Information about 8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above? covers all topics & solutions for Teaching 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above?.
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8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above?, a detailed solution for 8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above? has been provided alongside types of 8. The function f(x)=|x| |x 1| |x 2| |x - 1| * x-2| is (a) continuous, xin R (b) differentiable, xin R (c) only differentiable at x=0,-1-2 12 (d) None of the above? theory, EduRev gives you an
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