IIT JAM Exam > IIT JAM Questions > The function f(x) = |x|+|x - 1| isa)Continuou...
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The function f(x) = |x|+|x - 1| is
- a)Continuous at , but not differentiable at
- b)Both continuous and differentiable at x = 1
- c)Not continuous x = 1
- d)Not differentiable at x = 1
Correct answer is option 'A,D'. Can you explain this answer?
Verified Answer
The function f(x) = |x|+|x - 1| isa)Continuous at , but not differenti...
Given that f (x) = |x| + |x-1|, then f(1) = 1
Since absolute volue functions are continuous everywhere so f(x) = |x| + |x-1|. being the sum of two continuous function is continuous everywhere. Now we check differentiability at x = 1, we have
Hence Lf'(1) ≠ Rf'(1)
∴ Derivative do not exist at x = 1.
Since absolute volue functions are continuous everywhere so f(x) = |x| + |x-1|. being the sum of two continuous function is continuous everywhere. Now we check differentiability at x = 1, we have
Hence Lf'(1) ≠ Rf'(1)
∴ Derivative do not exist at x = 1.
Most Upvoted Answer
The function f(x) = |x|+|x - 1| isa)Continuous at , but not differenti...
Understanding the Function f(x)
The function in question is f(x) = |x| + |x - 1|. To analyze its properties, we need to consider the behavior of the absolute value functions involved.
Continuity of f(x)
- The function f(x) is composed of absolute value functions, which are continuous everywhere.
- Therefore, f(x) is continuous for all x, including at x = 1.
Checking Differentiability at x = 1
- To determine differentiability, we need to examine the left-hand and right-hand derivatives at x = 1.
Left-Hand Derivative
- For x < 1:="" />
f(x) = -x + (1 - x) = 1 - 2x
- The derivative, f'(x) = -2, when approaching from the left.
Right-Hand Derivative
- For x ≥ 1:
f(x) = x + (x - 1) = 2x - 1
- The derivative, f'(x) = 2, when approaching from the right.
Conclusion on Differentiability
- Since the left-hand derivative (-2) and right-hand derivative (2) are not equal at x = 1, f(x) is not differentiable at this point.
Summary
- Option A: The function is continuous at x = 1 but not differentiable at that point.
- Option D: The function is indeed not differentiable at x = 1 due to the differing slopes from either side.
Therefore, the correct assessment of the function f(x) = |x| + |x - 1| is that it is continuous at x = 1 but not differentiable there.
The function in question is f(x) = |x| + |x - 1|. To analyze its properties, we need to consider the behavior of the absolute value functions involved.
Continuity of f(x)
- The function f(x) is composed of absolute value functions, which are continuous everywhere.
- Therefore, f(x) is continuous for all x, including at x = 1.
Checking Differentiability at x = 1
- To determine differentiability, we need to examine the left-hand and right-hand derivatives at x = 1.
Left-Hand Derivative
- For x < 1:="" />
f(x) = -x + (1 - x) = 1 - 2x
- The derivative, f'(x) = -2, when approaching from the left.
Right-Hand Derivative
- For x ≥ 1:
f(x) = x + (x - 1) = 2x - 1
- The derivative, f'(x) = 2, when approaching from the right.
Conclusion on Differentiability
- Since the left-hand derivative (-2) and right-hand derivative (2) are not equal at x = 1, f(x) is not differentiable at this point.
Summary
- Option A: The function is continuous at x = 1 but not differentiable at that point.
- Option D: The function is indeed not differentiable at x = 1 due to the differing slopes from either side.
Therefore, the correct assessment of the function f(x) = |x| + |x - 1| is that it is continuous at x = 1 but not differentiable there.
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The function f(x) = |x|+|x - 1| isa)Continuous at , but not differenti...
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The function f(x) = |x|+|x - 1| isa)Continuous at , but not differentiable atb)Both continuous and differentiable at x = 1c)Not continuous x = 1d)Not differentiable at x = 1Correct answer is option 'A,D'. Can you explain this answer? for IIT JAM 2025 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The function f(x) = |x|+|x - 1| isa)Continuous at , but not differentiable atb)Both continuous and differentiable at x = 1c)Not continuous x = 1d)Not differentiable at x = 1Correct answer is option 'A,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function f(x) = |x|+|x - 1| isa)Continuous at , but not differentiable atb)Both continuous and differentiable at x = 1c)Not continuous x = 1d)Not differentiable at x = 1Correct answer is option 'A,D'. Can you explain this answer?.
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