f (x) = xx has a stationary point at
(A) x = e (B) x = 1/e
(C) x = 1 (D) x = √e
Last updated at Nov. 23, 2021 by Teachoo
Transcript
Question 29 f (x) = xx has a stationary point at (A) x = e (B) x = 1/๐ (C) x = 1 (D) x = โ๐ A stationary point of a function is a point where ๐โฒ(๐) = 0 For differentiating f (๐ฅ), we use logarithmic differentiation f (๐ฅ) = ๐ฅ^๐ฅ Taking log on both sides log f (๐) = ๐ log ๐ Differentiating w.r.t. x 1/๐(๐ฅ) ๐โฒ(๐ฅ) = ๐ฅ . 1/๐ฅ + 1. log ๐ฅ 1/๐ฅ^๐ฅ ๐โฒ(๐ฅ) = 1 + log ๐ฅ ๐โฒ(๐) = ๐^๐(1 + log ๐) Putting ๐โ(x) = 0 ๐ฅ^๐ฅ ("1 + log " ๐ฅ" " )=๐ Either ๐^๐ = 0 Since, ๐ฅ^๐ฅ is exponential function it can never be zero. Or 1 + log ๐ = 0 log ๐ฅ = โ1 Taking exponential on both sides ๐^๐๐๐โก๐ = ๐^(โ๐) ๐ฅ = ๐^(โ1) ๐ = ๐/๐ Hence, Stationary point is ๐ = ๐/๐ So, the correct answer is (B)
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