JEE Exam > JEE Questions > Given that f (x) =x1/x,x>0,has the maximu... Start Learning for Free
Given that f (x) = x1/x , x>0, has the maximum value at x = e,then
- a)eπ=πe
- b)eπ<πe
- c)eπ>πe
- d)eπ⩽πe
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)e&...
Most Upvoted Answer
Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)e&...
Understanding the Function
The function f(x) = x^(1/x) is given, and we need to find its maximum value.
Finding the Maximum Value
To find the maximum, we take the natural logarithm of f(x):
- Let y = f(x) = x^(1/x)
- Then, ln(y) = (1/x) * ln(x)
Next, we differentiate ln(y) with respect to x and set the derivative to zero:
- d/dx[ln(y)] = d/dx[(1/x) * ln(x)]
- Using quotient rule and simplifying leads to critical points.
Setting the Derivative to Zero
After simplification, we find the critical points by solving:
- 1 - ln(x) = 0
- Hence, ln(x) = 1
- This gives us x = e.
Verifying Maximum at x = e
To confirm that x = e is a maximum, we can use the second derivative test or analyze the behavior of the function around e. The function f(x) increases until x = e and then decreases afterwards.
Conclusion
Since the maximum value of the function f(x) occurs at x = e, we conclude:
- The correct answer is c) e.
This shows that the maximum value of the function f(x) = x^(1/x) indeed occurs at x = e, making option 'C' the correct choice.
The function f(x) = x^(1/x) is given, and we need to find its maximum value.
Finding the Maximum Value
To find the maximum, we take the natural logarithm of f(x):
- Let y = f(x) = x^(1/x)
- Then, ln(y) = (1/x) * ln(x)
Next, we differentiate ln(y) with respect to x and set the derivative to zero:
- d/dx[ln(y)] = d/dx[(1/x) * ln(x)]
- Using quotient rule and simplifying leads to critical points.
Setting the Derivative to Zero
After simplification, we find the critical points by solving:
- 1 - ln(x) = 0
- Hence, ln(x) = 1
- This gives us x = e.
Verifying Maximum at x = e
To confirm that x = e is a maximum, we can use the second derivative test or analyze the behavior of the function around e. The function f(x) increases until x = e and then decreases afterwards.
Conclusion
Since the maximum value of the function f(x) occurs at x = e, we conclude:
- The correct answer is c) e.
This shows that the maximum value of the function f(x) = x^(1/x) indeed occurs at x = e, making option 'C' the correct choice.
 | Explore Courses for JEE exam |  |
Top Courses for JEEView all
Top Courses for JEE
Question Description
Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer?.
Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice Given that f (x) =x1/x,x>0,has the maximum value at x = e,thena)eπ=πeb)eπ<πec)eπ>πed)eπ⩽πeCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice JEE tests.
| Explore Courses for JEE exam | |
Top Courses for JEE
Explore Courses
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
