A-question-related-to-Q-15184-Find-the-maximum-of-f-x-ln-x-1-x-

Question Number 15234 by mrW1 last updated on 08/Jun/17

A question related to Q . 15184

Find the maximum of f (x) = {(\ln x)}^{\frac{1}{x}}

Commented by mrW1 last updated on 09/Jun/17

the domain of f (x) = {(\ln x)}^{\frac{1}{x}} is [1, + \infty)

let t = \frac{1}{x}, t \in (0, 1]

y = {(\ln \frac{1}{t})}^{t} = {(- \ln t)}^{t}

\ln y = tln (- \ln t)

\frac{1}{y} \times \frac{dy}{dt} = \ln (- \ln t) + t \times \frac{1}{- \ln t} \times \frac{- 1}{t} = \ln (- \ln t) + \frac{1}{\ln t}

\frac{dy}{dt} = {(- \ln t)}^{t} [\ln (- \ln t) + \frac{1}{\ln t}] = 0

\Rightarrow \ln (- \ln t) + \frac{1}{\ln t} = 0

\Rightarrow t = 0.1714907907 (through graph)

\Rightarrow y_{\max} = 1.102147

Commented by mrW1 last updated on 09/Jun/17

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