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Question Number 133180 by mathlove last updated on 19/Feb/21

Commented by mr W last updated on 19/Feb/21

$n o r e a l s o l u t i o n!$ $x^{x} ⩾ \frac{1}{\sqrt[e]{e}} \approx 0.692$ $\frac{1}{256} << 0.692$ $\Rightarrow x^{x} = \frac{1}{256} i s n o t p o s s i b l e!$
Commented by mr W last updated on 20/Feb/21

$g e n e r a l l y$ $t h e e q u a t i o n x^{x} = a h a s$ $∙ n o s o l u t i o n i f a < \frac{1}{\sqrt[e]{e}} \approx 0.692$ $∙ o n e s o l u t i o n i f a = \frac{1}{\sqrt[e]{e}}$ $∙ t w o s o l u t i o n s i f \frac{1}{\sqrt[e]{e}} < a < 1$ $∙ o n e s o l u t i o n i f a ⩾ 1$ $t h e s o l u t i o n (s) i s (a r e) :$ $x = e^{W (\ln a)}$
Commented by Tawa11 last updated on 23/Jul/21

$great$
	Answered by JDamian last updated on 19/Feb/21

	$x = - 4$
	Commented by liberty last updated on 20/Feb/21

	Commented by mr W last updated on 20/Feb/21

	$4^{- 4} = \frac{1}{256}$ $b u t {(- 4)}^{(- 4)} i s n o t v a l i d, s o x = - 4 i s$ $n o t c o r r e c t .$
	Commented by liberty last updated on 20/Feb/21

	$in this graph clearly x^{x} = \frac{1}{256}$ $no has solution$
	Commented by JDamian last updated on 20/Feb/21

	$T h e q u e s t i o n i s n o t a b o u t h o w t h e$ $f u n c t i o n x^{x} a n d i t s g r a p h a r e .$ $I t i s a b o u t w h i c h v a l u e o f x p o w e r e d$ $t o i t s e l f e q u a l s \frac{1}{256} . N o c o n s t r a i n t s$ $a b o u t x v a l u e s a r e g i v e n .$ $Y o u a s s u m e d x ⩾ 0 a n d s o f o r g o t t o d r a w$ $t h e p o i n t (- 2, \frac{1}{4}) b e c a u s e {(- 2)}^{- 2} = \frac{1}{4}$ $i n t h e g r a p h .$
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