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The maximum value of (1/x)x is:
- a)e
- b)ee
- c)e1/e
- d)(1/e)1/e
Correct answer is option 'C'. Can you explain this answer?
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The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is...
Understanding the Function
To find the maximum value of the function (1/x)^x, we can rewrite it in a more manageable form.
Step 1: Rewrite the Function
The function can be expressed as:
f(x) = (1/x)^x = e^(x * ln(1/x)) = e^(-x * ln(x))
This transformation helps us analyze the function more easily.
Step 2: Differentiate the Function
To find the maximum point, we differentiate the function:
g(x) = -x * ln(x)
Now we differentiate g(x):
g'(x) = -ln(x) - 1
Set g'(x) = 0 to find critical points:
-ln(x) - 1 = 0
ln(x) = -1
x = e^(-1) = 1/e
Step 3: Determine the Maximum Value
Now, we must evaluate the second derivative or check the behavior of the first derivative around x = 1/e to confirm it is a maximum.
Calculating f(1/e):
f(1/e) = (1/(1/e))^(1/e) = e^(1/e)
Thus, the maximum value of the function (1/x)^x occurs at x = 1/e and is equal to e^(1/e).
Final Answer
The maximum value of (1/x)^x is:
e^(1/e)
Therefore, the correct answer is option 'C'.
To find the maximum value of the function (1/x)^x, we can rewrite it in a more manageable form.
Step 1: Rewrite the Function
The function can be expressed as:
f(x) = (1/x)^x = e^(x * ln(1/x)) = e^(-x * ln(x))
This transformation helps us analyze the function more easily.
Step 2: Differentiate the Function
To find the maximum point, we differentiate the function:
g(x) = -x * ln(x)
Now we differentiate g(x):
g'(x) = -ln(x) - 1
Set g'(x) = 0 to find critical points:
-ln(x) - 1 = 0
ln(x) = -1
x = e^(-1) = 1/e
Step 3: Determine the Maximum Value
Now, we must evaluate the second derivative or check the behavior of the first derivative around x = 1/e to confirm it is a maximum.
Calculating f(1/e):
f(1/e) = (1/(1/e))^(1/e) = e^(1/e)
Thus, the maximum value of the function (1/x)^x occurs at x = 1/e and is equal to e^(1/e).
Final Answer
The maximum value of (1/x)^x is:
e^(1/e)
Therefore, the correct answer is option 'C'.
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The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is option 'C'. Can you explain this answer?.
The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The maximum value of (1/x)xis:a)eb)eec)e1/ed)(1/e)1/eCorrect answer is option 'C'. Can you explain this answer?.
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