f (x) = xx has a stationary point at

(A) x = e                      (B) x = 1/e

(C) x = 1                      (D) x = √e



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Question 16 f (x) = xx has a stationary point at (A) x = e (B) x = 1/𝑒 (C) x = 1 (D) x = √𝑒 A stationary point of a function is a point where 𝒇′(𝒙) = 0 For differentiating f (𝑥), we use logarithmic differentiation f (𝑥) = 𝑥^𝑥 Taking log on both sides log f (𝒙) = 𝒙 log 𝒙 Differentiating w.r.t. x 1/𝑓(𝑥) 𝑓′(𝑥) = 𝑥 . 1/𝑥 + 1. log 𝑥 1/𝑥^𝑥 𝑓′(𝑥) = 1 + log 𝑥 𝒇′(𝒙) = 𝒙^𝒙(1 + log 𝒙) Putting 𝒇’(x) = 0 𝑥^𝑥 ("1 + log " 𝑥" " )=𝟎 Either 𝒙^𝒙 = 0 Since, 𝑥^𝑥 is exponential function it can never be zero. Or 1 + log 𝒙 = 0 log 𝑥 = −1 Taking exponential on both sides 𝒆^𝒍𝒐𝒈⁡𝒙 = 𝒆^(−𝟏) 𝑥 = 𝑒^(−1) 𝒙 = 𝟏/𝒆 Hence, Stationary point is 𝒙 = 𝟏/𝒆 So, the correct answer is (B)

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.