The maximum value of (1/x) ^{ x } is:
(A) e (B) ee
(C) e ^{ (1/e) } (D) 1/e ^{ (1/e) }
Last updated at Nov. 23, 2021 by Teachoo
Transcript
Question 30 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)
NCERT Exemplar - MCQs
Question 2
Question 3 Important
Question 4
Question 5
Question 6 Important
Question 7 Important
Question 8
Question 9 Important
Question 10
Question 11 Important
Question 12 Important
Question 13
Question 14
Question 15
Question 16 Important
Question 17 Important
Question 18 Important
Question 19
Question 20 Important
Question 21 Important
Question 22 Important
Question 23
Question 24 Important
Question 25 Important
Question 26
Question 27
Question 28 Important
Question 29 Important
Question 30 Important You are here
NCERT Exemplar - MCQs
About the Author