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If Rolle's theorem holds for f(x) = x3 − 6x2 + kx + 5 on [1, 3] with c = +1/3 , then the value of k is _____.(Answer up to the nearest integer)
Correct answer is '11'. Can you explain this answer?
Most Upvoted Answer
If Rolles theorem holds for f(x) = x3 − 6x2 + kx + 5 on [1, 3] w...
⇒144 - 12k = 12 ⇒k = 11
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If Rolles theorem holds for f(x) = x3 − 6x2 + kx + 5 on [1, 3] w...
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 Details
For the given function f(x) = x³ - 6x² + kx + 5:
- Interval: [1, 3]
- c: Given as +1/3
Step 1: Check Continuity and Differentiability
- The function is a polynomial, thus continuous and differentiable everywhere, including the interval [1, 3].
Step 2: Calculate f(1) and f(3)
- f(1):
f(1) = 1³ - 6(1)² + k(1) + 5 = 1 - 6 + k + 5 = k
- f(3):
f(3) = 3³ - 6(3)² + k(3) + 5 = 27 - 54 + 3k + 5 = 3k - 22
Step 3: Set f(1) = f(3)
To satisfy Rolle's theorem:
- k = 3k - 22
Rearranging gives:
- 22 = 2k
- k = 11
Conclusion
Thus, the value of k, satisfying the conditions of Rolle's theorem for the function on the interval [1, 3] is:
- k = 11
This aligns with the correct answer provided.
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 Details
For the given function f(x) = x³ - 6x² + kx + 5:
- Interval: [1, 3]
- c: Given as +1/3
Step 1: Check Continuity and Differentiability
- The function is a polynomial, thus continuous and differentiable everywhere, including the interval [1, 3].
Step 2: Calculate f(1) and f(3)
- f(1):
f(1) = 1³ - 6(1)² + k(1) + 5 = 1 - 6 + k + 5 = k
- f(3):
f(3) = 3³ - 6(3)² + k(3) + 5 = 27 - 54 + 3k + 5 = 3k - 22
Step 3: Set f(1) = f(3)
To satisfy Rolle's theorem:
- k = 3k - 22
Rearranging gives:
- 22 = 2k
- k = 11
Conclusion
Thus, the value of k, satisfying the conditions of Rolle's theorem for the function on the interval [1, 3] is:
- k = 11
This aligns with the correct answer provided.
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Question Description
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