Functions
Given f(x)=x1/x for x>0. The maximum value of f(x) is:
x^(1/x) maximum
Select the correct option:
Solution
e1/e
-
Take logs: y=x1/x⟹lny=xlnx.
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Differentiate: y1y′=x21⋅x−lnx⋅1=x21−lnx.
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Find Critical Point: y′=0⟹1−lnx=0⟹lnx=1⟹x=e.
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Check sign change: For x<e,1−lnx>0 (Increasing). For x>e,1−lnx<0 (Decreasing). So Max occurs at x=e.
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Max Value: f(e)=e1/e.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- functions
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
e1/e
-
Take logs: y=x1/x⟹lny=xlnx.
-
Differentiate: y1y′=x21⋅x−lnx⋅1=x21−lnx.
-
Find Critical Point: y′=0⟹1−lnx=0⟹lnx=1⟹x=e.
-
Check sign change: For x<e,1−lnx>0 (Increasing). For x>e,1−lnx<0 (Decreasing). So Max occurs at x=e.
-
Max Value: f(e)=e1/e.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of functions. It appeared in the 2025 exam.
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