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Consider the function f ( x ) = | x - 1 |+ x2 , w here x ∈R .
Which one of the following statements is correct?
- a)f(x) is continuous but not differentiable at x = 0
- b)f(x) is continuous but not differentiable at x = 1
- c)f(x) is differentiable at x = 1
- d)f(x) is not differentiable at x = 0 and x = 1
Correct answer is option 'B'. Can you explain this answer?
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Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which...
f(x) = | x - l | + x2 
x ∈R
f1 (x ) = |x - l |, f2(x) = x2
f1 (x) and f2{x) both are continuous.
Hence f(x) is continuous.
f(x) in differentiable at x = 0
f1(x) is not differentiable at x = 1.
Hence(fx) is continuous but not differentiable at x= 1
x ∈Rf1 (x ) = |x - l |, f2(x) = x2
f1 (x) and f2{x) both are continuous.
Hence f(x) is continuous.
f(x) in differentiable at x = 0
f1(x) is not differentiable at x = 1.
Hence(fx) is continuous but not differentiable at x= 1
Most Upvoted Answer
Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which...
Explanation:
Definition of Continuity and Differentiability:
- A function f(x) is said to be continuous at a point x = c if the limit of f(x) as x approaches c exists and is equal to f(c).
- A function f(x) is said to be differentiable at a point x = c if the derivative of f(x) exists at x = c.
Given Function:
f(x) = |x - 1| + x^2
Analysis:
- At x = 1, the function f(x) can be rewritten as f(x) = |1 - 1| + 1^2 = 1, which is a continuous function at x = 1.
- At x = 0, the function f(x) can be rewritten as f(x) = |0 - 1| + 0^2 = 1, which is also a continuous function at x = 0.
- However, when checking the differentiability at these points, we need to consider the behavior of the function as it approaches these points.
Explanation of Correct Answer:
- The function f(x) is continuous but not differentiable at x = 1, as the function |x - 1| has a sharp point at x = 1 where the function is not smooth.
- At x = 0, the function f(x) is also continuous but not differentiable, as the function |x - 1| has a sharp point at x = 0 as well.
Therefore, option B, "f(x) is continuous but not differentiable at x = 1," is the correct statement based on the given function f(x) = |x - 1| + x^2.
Definition of Continuity and Differentiability:
- A function f(x) is said to be continuous at a point x = c if the limit of f(x) as x approaches c exists and is equal to f(c).
- A function f(x) is said to be differentiable at a point x = c if the derivative of f(x) exists at x = c.
Given Function:
f(x) = |x - 1| + x^2
Analysis:
- At x = 1, the function f(x) can be rewritten as f(x) = |1 - 1| + 1^2 = 1, which is a continuous function at x = 1.
- At x = 0, the function f(x) can be rewritten as f(x) = |0 - 1| + 0^2 = 1, which is also a continuous function at x = 0.
- However, when checking the differentiability at these points, we need to consider the behavior of the function as it approaches these points.
Explanation of Correct Answer:
- The function f(x) is continuous but not differentiable at x = 1, as the function |x - 1| has a sharp point at x = 1 where the function is not smooth.
- At x = 0, the function f(x) is also continuous but not differentiable, as the function |x - 1| has a sharp point at x = 0 as well.
Therefore, option B, "f(x) is continuous but not differentiable at x = 1," is the correct statement based on the given function f(x) = |x - 1| + x^2.
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Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer?
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Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer?.
Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer?.
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Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer?, a detailed solution for Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of Consider the function f ( x ) = | x - 1 |+ x2 , w here x∈R .Which one of the following statements is correct?a)f(x) is continuous but not differentiable at x = 0b)f(x) is continuous but not differentiable at x = 1c)f(x) is differentiable at x = 1d)f(x) is not differentiable at x = 0 and x = 1Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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