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If f(x) = xα In x and f(0) = 0, then the value of α for which Rolle’s theorem can be applied in [0,1] is,
- a)-2
- b)-1
- c)0
- d)1/2
Correct answer is option 'D'. Can you explain this answer?
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The function has to be continuous in [0,1]
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Understanding Rolle's Theorem
Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.
Function Definition
Given the function:
- f(x) = x^α * ln(x)
- f(0) = 0 (ensured by taking the limit as x approaches 0)
Conditions for Rolle's Theorem
To apply Rolle's theorem on the interval [0, 1], we must ensure:
1. Continuity at [0, 1]:
- The function must be continuous at both endpoints.
- As x approaches 0, f(x) approaches 0 if α > -1 (since ln(x) approaches -∞).
2. Differentiability on (0, 1):
- The function must be differentiable within the interval (0, 1). The term x^α * ln(x) is differentiable if α > -1.
3. Equal Values:
- We need f(0) = f(1).
- Evaluating f(1): f(1) = 1^α * ln(1) = 0.
- Therefore, we need f(0) = 0, which is already given.
Finding the Suitable α
To ensure f(x) remains defined and continuous at x = 0:
- The limit of f(x) as x approaches 0 must equal 0.
- This occurs if α > -1.
However, to ensure the function remains valid and differentiable, we check:
- For α = -1, f(x) = ln(x) which diverges.
- For α = -2, the function becomes undefined.
- For α = 0, f(x) approaches 0 correctly.
- For α = 1/2, f(0) = 0 and is continuous and differentiable.
Conclusion
Thus, the correct value of α for which Rolle's theorem can be applied in [0, 1] is indeed:
- α = 1/2 (Option D)
Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.
Function Definition
Given the function:
- f(x) = x^α * ln(x)
- f(0) = 0 (ensured by taking the limit as x approaches 0)
Conditions for Rolle's Theorem
To apply Rolle's theorem on the interval [0, 1], we must ensure:
1. Continuity at [0, 1]:
- The function must be continuous at both endpoints.
- As x approaches 0, f(x) approaches 0 if α > -1 (since ln(x) approaches -∞).
2. Differentiability on (0, 1):
- The function must be differentiable within the interval (0, 1). The term x^α * ln(x) is differentiable if α > -1.
3. Equal Values:
- We need f(0) = f(1).
- Evaluating f(1): f(1) = 1^α * ln(1) = 0.
- Therefore, we need f(0) = 0, which is already given.
Finding the Suitable α
To ensure f(x) remains defined and continuous at x = 0:
- The limit of f(x) as x approaches 0 must equal 0.
- This occurs if α > -1.
However, to ensure the function remains valid and differentiable, we check:
- For α = -1, f(x) = ln(x) which diverges.
- For α = -2, the function becomes undefined.
- For α = 0, f(x) approaches 0 correctly.
- For α = 1/2, f(0) = 0 and is continuous and differentiable.
Conclusion
Thus, the correct value of α for which Rolle's theorem can be applied in [0, 1] is indeed:
- α = 1/2 (Option D)
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