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.1x−1x2logex
or f′(x)=x1/xx2[1−logex] ...(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)eπ⇒πe<eπ.
Thus eπ is bigger that πe.