find-the-minimum-value-of-f-x-x-x-for-x-R-

Question Number 96586 by M±th+et+s last updated on 02/Jun/20

f i n d t h e m i n i m u m v a l u e o f

f (x) = x^{x}

f o r x \in R^{+}

Answered by 1549442205 last updated on 02/Jun/20

y = x^{x} \Rightarrow lny = xlnx \Rightarrow \frac{y^{'}}{y} = lnx + 1

\Rightarrow y^{'} = x^{x} (lnx + 1) = 0 \Leftrightarrow lnx = - 1

\Leftrightarrow x = e^{- 1} . y^{″} = \frac{x^{x}}{x} + (lnx + 1) . x^{x} (lnx + 1)

= x^{x} (\frac{1}{x} + {(lnx + 1)}^{2}) . y^{″} (e^{- 1}) =

e^{- e^{- 1}} . e = e^{1 - e^{- 1}} > 0 . Hence

miny = minf (x) = f (e^{- 1}) = e^{- e^{- 1}}

Commented by M±th+et+s last updated on 03/Jun/20

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