solve-for-x-e-x-e-x-2-e-x-3-3-x-x-2-x-3-

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 \dots 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 \dots a l g e b r i c \dots i n t e r s e c t \dots 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