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Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. Then n can be any element of the interval
- a)[1, ∞)
- b)(0, ∞)
- c)(1/2, ∞)
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. ...
Calculation:
Given: f(x) = xn, n ≠ 0.
F’(x) = nxn – 1
For f(x) to be differentiable for all values of x, n – 1 ≥ 0
So, n ≥ 1.
Given: f(x) = xn, n ≠ 0.
F’(x) = nxn – 1
For f(x) to be differentiable for all values of x, n – 1 ≥ 0
So, n ≥ 1.
Most Upvoted Answer
Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. ...
Understanding the Function f(x) = x^n
The function f(x) = x^n, where n ≠ 0, is a power function. To determine the conditions under which this function is differentiable, we need to analyze the nature of the exponent n.
1. Differentiability Over the Domain
- A function is differentiable at a point if it has a defined derivative at that point.
- For f(x) = x^n, it is differentiable for all x > 0 when n is any real number.
2. Analyzing the Exponent n
- If n is in the interval [1, ∞), the function grows positively and is smooth (no corners or cusps) for all x > 0.
- If n is in (0, ∞), the function remains differentiable for all positive x.
- If n is less than 0, particularly when it approaches x = 0, the function can become undefined or non-differentiable.
3. Behavior at x = 0
- For n < 0,="" f(x)="x^n" approaches="" infinity="" as="" x="" approaches="" 0,="" making="" it="" non-differentiable="" at="" x="" />
- Hence, for the function to be differentiable at all x, n must be at least 1.
Conclusion
- The only valid range for n to ensure f(x) is differentiable for all x (especially at x = 0) is n in the interval [1, ∞).
- Therefore, the correct answer is option 'a) [1, ∞)'.
This analysis highlights the importance of the exponent n in determining the differentiability of the power function across its domain.
The function f(x) = x^n, where n ≠ 0, is a power function. To determine the conditions under which this function is differentiable, we need to analyze the nature of the exponent n.
1. Differentiability Over the Domain
- A function is differentiable at a point if it has a defined derivative at that point.
- For f(x) = x^n, it is differentiable for all x > 0 when n is any real number.
2. Analyzing the Exponent n
- If n is in the interval [1, ∞), the function grows positively and is smooth (no corners or cusps) for all x > 0.
- If n is in (0, ∞), the function remains differentiable for all positive x.
- If n is less than 0, particularly when it approaches x = 0, the function can become undefined or non-differentiable.
3. Behavior at x = 0
- For n < 0,="" f(x)="x^n" approaches="" infinity="" as="" x="" approaches="" 0,="" making="" it="" non-differentiable="" at="" x="" />
- Hence, for the function to be differentiable at all x, n must be at least 1.
Conclusion
- The only valid range for n to ensure f(x) is differentiable for all x (especially at x = 0) is n in the interval [1, ∞).
- Therefore, the correct answer is option 'a) [1, ∞)'.
This analysis highlights the importance of the exponent n in determining the differentiability of the power function across its domain.
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Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. Then n can be any element of the intervala)[1, ∞)b)(0, ∞)c)(1/2,∞)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. Then n can be any element of the intervala)[1, ∞)b)(0, ∞)c)(1/2,∞)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. Then n can be any element of the intervala)[1, ∞)b)(0, ∞)c)(1/2,∞)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose the function f(x) = xn, n ≠ 0 is differentiable for all x. Then n can be any element of the intervala)[1, ∞)b)(0, ∞)c)(1/2,∞)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer?.
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