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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  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

Transcript

Question 29 f (x) = xx has a stationary point at (A) x = e (B) x = 1/๐‘’ (C) x = 1 (D) x = โˆš๐‘’ A stationary point of a function is a point where ๐’‡โ€ฒ(๐’™) = 0 For differentiating f (๐‘ฅ), we use logarithmic differentiation f (๐‘ฅ) = ๐‘ฅ^๐‘ฅ Taking log on both sides log f (๐’™) = ๐’™ log ๐’™ Differentiating w.r.t. x 1/๐‘“(๐‘ฅ) ๐‘“โ€ฒ(๐‘ฅ) = ๐‘ฅ . 1/๐‘ฅ + 1. log ๐‘ฅ 1/๐‘ฅ^๐‘ฅ ๐‘“โ€ฒ(๐‘ฅ) = 1 + log ๐‘ฅ ๐’‡โ€ฒ(๐’™) = ๐’™^๐’™(1 + log ๐’™) Putting ๐’‡โ€™(x) = 0 ๐‘ฅ^๐‘ฅ ("1 + log " ๐‘ฅ" " )=๐ŸŽ Either ๐’™^๐’™ = 0 Since, ๐‘ฅ^๐‘ฅ is exponential function it can never be zero. Or 1 + log ๐’™ = 0 log ๐‘ฅ = โˆ’1 Taking exponential on both sides ๐’†^๐’๐’๐’ˆโก๐’™ = ๐’†^(โˆ’๐Ÿ) ๐‘ฅ = ๐‘’^(โˆ’1) ๐’™ = ๐Ÿ/๐’† Hence, Stationary point is ๐’™ = ๐Ÿ/๐’† 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 10 years. He provides courses for Maths and Science at Teachoo.