if-y-x-x-what-is-the-range-of-this-function-

Question Number 17065 by mrW1 last updated on 30/Jun/17

if y = x^{x}

what is the range of this function ?

Commented by ajfour last updated on 30/Jun/17

y \in [e^{- 1 / e}, \infty) .

Commented by mrW1 last updated on 30/Jun/17

{that}^{'} s right .

Commented by ajfour last updated on 30/Jun/17

Sir, i evaluated \lim_{x \to 0} x^{x} taking

logarithm, got 1 . Then found

\frac{dy}{dx}, saw it is - ve at the start and

zero for x = \frac{1}{e}, and is + ve thereafter .

so the minimum of x^{x} = {(\frac{1}{e})}^{1 / e} .

Commented by mrW1 last updated on 30/Jun/17

I would also do like this and find the

minimum of the function which is

{(\frac{1}{e})}^{\frac{1}{e}} = e^{- \frac{1}{e}} = \frac{1}{^{e} \sqrt{e}}

But there is also an alternative way .

y = x^{x}

the inverse function is

x = \frac{\ln y}{W (\ln y)} with W = lambert function

its domain is - \frac{1}{e} ⩽ \ln y < \infty or

e^{- \frac{1}{e}} ⩽ y < \infty

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