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 = 1Β Β Β Β Β Β Β Β Β Β Β (D) x = βe
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Last updated at March 29, 2023 by Teachoo
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Question 29 f (x) = xx has a stationary point at (A) x = e (B) x = 1/π (C) x = 1 (D) x = βπ A stationary point of a function is a point where πβ²(π) = 0 For differentiating f (π₯), we use logarithmic differentiation f (π₯) = π₯^π₯ Taking log on both sides log f (π) = π log π Differentiating w.r.t. x 1/π(π₯) πβ²(π₯) = π₯ . 1/π₯ + 1. log π₯ 1/π₯^π₯ πβ²(π₯) = 1 + log π₯ πβ²(π) = π^π(1 + log π) Putting πβ(x) = 0 π₯^π₯ ("1 + log " π₯" " )=π Either π^π = 0 Since, π₯^π₯ is exponential function it can never be zero. Or 1 + log π = 0 log π₯ = β1 Taking exponential on both sides π^πππβ‘π = π^(βπ) π₯ = π^(β1) π = π/π Hence, Stationary point is π = π/π So, the correct answer is (B)