##
The maximum value of (1/x)
^{
x
}
is:

## (A) e (B) ee

##
(C) e
^{
(1/e)
}
(D) 1/e
^{
(1/e)
}

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Chapter 6 Class 12 Application of Derivatives

Serial order wise

Last updated at April 16, 2024 by Teachoo

Question 17 The maximum value of (1/๐ฅ)^๐ฅ is: (A) e (B) ee (C) ๐^(1/๐) (D) ใ1/๐ใ^(1/๐) Let f (๐ฅ) = (1/๐ฅ)^๐ฅ To find maximum value, we need to differentiate f(x) For differentiating f (๐ฅ), we use logarithmic differentiation f (๐ฅ) = (1/๐ฅ)^๐ฅ log (f(x)) = ๐ log (๐/๐) Differentiating wrt ๐ฅ ๐/(๐(๐)) fโ(x) = 1โlog (๐/๐) + ๐ ร (๐/(๐/๐)) ร ((โ๐)/๐^๐ ) 1/(๐(๐ฅ)) fโ(x) = log (1/๐ฅ) + ๐ฅ ร (๐ฅ) ร ((โ1)/๐ฅ^2 ) 1/(๐(๐ฅ)) fโ(x) = log (1/๐ฅ) + ๐ฅ^2 ร ((โ1)/๐ฅ^2 ) 1/(๐(๐ฅ)) fโ(x) = log (1/๐ฅ) โ 1 fโ(x) = f(x) [logโกใ(1/๐ฅ)โ1ใ ] Putting f (๐ฅ) =(1/๐ฅ)^๐ฅ fโ(x) = (๐/๐)^๐ (๐ฅ๐จ๐ โกใ(๐/๐)โ๐ใ ) Putting fโ(x) = 0 (1/๐ฅ)^๐ฅ (logโกใ(1/๐ฅ)โ1ใ ) = 0 Since, there is only one critical point, so it will be point of maxima Either (๐/๐)^๐ = 0 Since, it is an exponential function It can never be zero. Or (๐๐๐โกใ(๐/๐)โ๐ใ ) = 0 log 1/๐ฅ = 1 Taking exponential on both sides e^logโกใ1/xใ = ๐^1 1/๐ฅ = e ๐ = ๐^(โ๐) Putting ๐ฅ = 1/๐ in f (x) f (๐/๐) = (1/(1/๐))^(1/๐) f (๐/๐) = ๐^(๐/๐) So, the correct answer is (C)