solve for x: e^x + e^x^2 + e^x^3 = 3 + x + x^2 + x^3


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Question Number 38587 by tawa tawa last updated on 27/Jun/18

$solve for x : e^{x} + e^{x^{2}} + e^{x^{3}} = 3 + x + x^{2} + x^{3}$
Commented by tanmay.chaudhury50@gmail.com last updated on 27/Jun/18

$x = 0$
Commented by tanmay.chaudhury50@gmail.com last updated on 27/Jun/18

$l e f t h a n d s i d e c u r v e . . . e x p o n e n t i a l a n d r i g h t$ $h a n d s i d e c u r v e . . . a l g e b r i c . . . i n t e r s e c t . . . a t o n e$ $p o i n t (0, 3)$
	Answered by tanmay.chaudhury50@gmail.com last updated on 27/Jun/18

	Answered by MrW3 last updated on 29/Jun/18

	${l e t}^{'} s h a v e a l o o k a t t h e f u n c t i o n$ $f (x) = e^{x} - x$ $f^{'} (x) = e^{x} - 1 = 0 \Rightarrow x = 0 \Rightarrow f (0) = 1$ $f^{″} (x) = e^{x} \Rightarrow f^{″} (0) = 1 > 0$ $t h a t m e a n s f (x) = e^{x} - x h a s i t s m i n i m u m$ $a t x = 0 a n d t h e m i n i m u m i s 1, o r$ $i n o t h e r w o r d s :$ $f (x) = e^{x} - x ⩾ 1$ $s i m i l a r l y e^{x^{2}} - x^{2} ⩾ 1, e^{x^{3}} - x^{3} ⩾ 1$ $\Rightarrow (e^{x} - x) + (e^{x^{2}} - x^{2}) + (e^{x^{3}} - x^{3}) ⩾ 3$ $\Rightarrow e^{x} + e^{x^{2}} + e^{x^{3}} ⩾ 3 + x + x^{2} + x^{3}$ $t h e = s i g n i s v a l i d o n l y a t x = 0 .$ $t h a t i s t o s a y$ $e^{x} + e^{x^{2}} + e^{x^{3}} = 3 + x + x^{2} + x^{3} h a s o n l y$ $o n e s o l u t i o n x = 0 .$
	Commented by tawa tawa last updated on 29/Jun/18

	$God bless you sir$
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