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If f (x) = log5 + log (x3 - 3), where x [-1, 1], then find the value of c by using Rolle's theorem.
- a)1
- b)-1
- c)0
- d)2
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If f (x) = log5 + log (x3 - 3), where x [-1, 1], then find the value o...
Given,
Then, differentiate this equation on both sides.
Thus, c = 0
Hence, this is the required solution.
Then, differentiate this equation on both sides.
Thus, c = 0
Hence, this is the required solution.
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If f (x) = log5 + log (x3 - 3), where x [-1, 1], then find the value o...
Explanation:
Rolle's Theorem:
Rolle's Theorem states that if a function f(x) is continuous on a closed interval [a, b] and 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.
Given Function:
f(x) = log5 + log (x^3 - 3)
Conditions for applying Rolle's Theorem:
- The function should be continuous on the closed interval [-1, 1]
- The function should be differentiable on the open interval (-1, 1)
- f(-1) = f(1)
Applying Rolle's Theorem:
Now, we need to check if the conditions for Rolle's Theorem are satisfied for the given function.
f(x) is continuous on the closed interval [-1, 1] as it is a sum of continuous functions.
f(x) is differentiable on the open interval (-1, 1) as it is a sum of differentiable functions.
f(-1) = log5 + log((-1)^3 - 3) = log5 + log(-4)
f(1) = log5 + log(1^3 - 3) = log5 + log(-2)
Since f(-1) is not equal to f(1), we cannot directly apply Rolle's Theorem to find the value of c.
Conclusion:
As the conditions for Rolle's Theorem are not satisfied for the given function, we cannot find the value of c using Rolle's Theorem in this case. Hence, the correct answer is option 'c) 0'.
Rolle's Theorem:
Rolle's Theorem states that if a function f(x) is continuous on a closed interval [a, b] and 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.
Given Function:
f(x) = log5 + log (x^3 - 3)
Conditions for applying Rolle's Theorem:
- The function should be continuous on the closed interval [-1, 1]
- The function should be differentiable on the open interval (-1, 1)
- f(-1) = f(1)
Applying Rolle's Theorem:
Now, we need to check if the conditions for Rolle's Theorem are satisfied for the given function.
f(x) is continuous on the closed interval [-1, 1] as it is a sum of continuous functions.
f(x) is differentiable on the open interval (-1, 1) as it is a sum of differentiable functions.
f(-1) = log5 + log((-1)^3 - 3) = log5 + log(-4)
f(1) = log5 + log(1^3 - 3) = log5 + log(-2)
Since f(-1) is not equal to f(1), we cannot directly apply Rolle's Theorem to find the value of c.
Conclusion:
As the conditions for Rolle's Theorem are not satisfied for the given function, we cannot find the value of c using Rolle's Theorem in this case. Hence, the correct answer is option 'c) 0'.
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If f (x) = log5 + log (x3 - 3), where x [-1, 1], then find the value of c by using Rolles theorem.a)1b)-1c)0d)2Correct answer is option 'C'. Can you explain this answer?
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