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Question

Assertion :eπ>πe Reason: The function f(x)=x1x attains global maxima at x = e.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
We have f(x)=x1/x
logf(x)=1xlogex,
Differentiating with respect to x, we get
1f(x).f(x)=1x.1x1x2logex
or f(x)=x1/xx2[1logex] ...(1)
We have to consider the function for x=e to x=π,π>e
for e<x<π,logex>1 as x>e.
Then it follows from
(1) that f(x)<0 for
x>e.
Hence f(x) is a decreasing function of x for x>e. Since π>e,
we conclude f(π)<f(e)π1/π<e1/e
(π1/π)eπ<(e1/e)eππe<eπ.
Thus eπ is bigger that πe.

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