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Maximum value of function f(x) = x/8 + 2/x on the interval [1, 6] is
- a)1
- b)9/8
- c)13/12
- d)17/8
Correct answer is option 'D'. Can you explain this answer?
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Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1...
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Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1...
Introduction
To determine the maximum value of the function f(x) = x/8 + 2/x on the interval [1, 6], we will analyze the function at the endpoints and find critical points within the interval.
Step 1: Find the Critical Points
1. Differentiate the function:
- f'(x) = 1/8 - 2/x²
2. Set the derivative to zero to find critical points:
- 1/8 - 2/x² = 0
- Solving gives:
- 2/x² = 1/8
- x² = 16
- x = 4 (Only positive value in the interval)
Step 2: Evaluate the Function at Endpoints and Critical Points
1. Calculate f(1):
- f(1) = 1/8 + 2/1 = 1/8 + 2 = 17/8
2. Calculate f(4):
- f(4) = 4/8 + 2/4 = 1/2 + 1/2 = 1
3. Calculate f(6):
- f(6) = 6/8 + 2/6 = 3/4 + 1/3
- Convert to a common denominator (12):
- f(6) = 9/12 + 4/12 = 13/12
Step 3: Compare Values
- f(1) = 17/8
- f(4) = 1
- f(6) = 13/12
Conclusion
The maximum value of f(x) on the interval [1, 6] is at x = 1, which gives f(1) = 17/8. Hence, the correct answer is option 'D'.
To determine the maximum value of the function f(x) = x/8 + 2/x on the interval [1, 6], we will analyze the function at the endpoints and find critical points within the interval.
Step 1: Find the Critical Points
1. Differentiate the function:
- f'(x) = 1/8 - 2/x²
2. Set the derivative to zero to find critical points:
- 1/8 - 2/x² = 0
- Solving gives:
- 2/x² = 1/8
- x² = 16
- x = 4 (Only positive value in the interval)
Step 2: Evaluate the Function at Endpoints and Critical Points
1. Calculate f(1):
- f(1) = 1/8 + 2/1 = 1/8 + 2 = 17/8
2. Calculate f(4):
- f(4) = 4/8 + 2/4 = 1/2 + 1/2 = 1
3. Calculate f(6):
- f(6) = 6/8 + 2/6 = 3/4 + 1/3
- Convert to a common denominator (12):
- f(6) = 9/12 + 4/12 = 13/12
Step 3: Compare Values
- f(1) = 17/8
- f(4) = 1
- f(6) = 13/12
Conclusion
The maximum value of f(x) on the interval [1, 6] is at x = 1, which gives f(1) = 17/8. Hence, the correct answer is option 'D'.
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Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1b)9/8c)13/12d)17/8Correct answer is option 'D'. 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 Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1b)9/8c)13/12d)17/8Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1b)9/8c)13/12d)17/8Correct answer is option 'D'. Can you explain this answer?.
Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1b)9/8c)13/12d)17/8Correct answer is option 'D'. 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 Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1b)9/8c)13/12d)17/8Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Maximum value of function f(x) = x/8 + 2/xon the interval [1, 6] isa)1b)9/8c)13/12d)17/8Correct answer is option 'D'. Can you explain this answer?.
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