Check sibling questions

f (x) = xx has a stationary point at

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

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

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Transcript

Question 29 f (x) = xx has a stationary point at (A) x = e (B) x = 1/𝑒 (C) x = 1 (D) x = βˆšπ‘’ A stationary point of a function is a point where 𝒇′(𝒙) = 0 For differentiating f (π‘₯), we use logarithmic differentiation f (π‘₯) = π‘₯^π‘₯ Taking log on both sides log f (𝒙) = 𝒙 log 𝒙 Differentiating w.r.t. x 1/𝑓(π‘₯) 𝑓′(π‘₯) = π‘₯ . 1/π‘₯ + 1. log π‘₯ 1/π‘₯^π‘₯ 𝑓′(π‘₯) = 1 + log π‘₯ 𝒇′(𝒙) = 𝒙^𝒙(1 + log 𝒙) Putting 𝒇’(x) = 0 π‘₯^π‘₯ ("1 + log " π‘₯" " )=𝟎 Either 𝒙^𝒙 = 0 Since, π‘₯^π‘₯ is exponential function it can never be zero. Or 1 + log 𝒙 = 0 log π‘₯ = βˆ’1 Taking exponential on both sides 𝒆^π’π’π’ˆβ‘π’™ = 𝒆^(βˆ’πŸ) π‘₯ = 𝑒^(βˆ’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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.