if x;y;z>0 prove that... ((x/z))^2 e^(((z/x))^2 ) + ((y/x))^2 e^(((x/y))^2 ) + ((z/y))^2 e^(((y/z))^2 ) ≥ 3e


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 143410 by mathdanisur last updated on 14/Jun/21

$i f x; y; z > 0 p r o v e t h a t . . .$ ${(\frac{x}{z})}^{2} e^{{(\frac{z}{x})}^{2}} + {(\frac{y}{x})}^{2} e^{{(\frac{x}{y})}^{2}} + {(\frac{z}{y})}^{2} e^{{(\frac{y}{z})}^{2}} ⩾ 3 e$
Commented bymr W last updated on 14/Jun/21

$u s i n g$ $f (x) = \frac{e^{x}}{x} ⩾ e f o r x > 0$
Commented bymathdanisur last updated on 14/Jun/21

$S i r, e x a c t l y h o w i s t h e s o l u t i o n p l e a s e$
Commented bymr W last updated on 14/Jun/21

$f (x) = \frac{e^{x}}{x}$ $f^{'} (x) = \frac{e^{x}}{x} - \frac{e^{x}}{x^{2}} = 0 \Rightarrow x = 1$ $f^{″} (x) = \frac{e^{x}}{x} - \frac{2 e^{x}}{x^{2}} + \frac{2 e^{x}}{x^{3}}$ $f^{″} (1) = e > 0$ $f (1) = e i s m i n i m u m o f f (x), i . e .$ $f o r x > 0 : \frac{e^{x}}{x} ⩾ e .$ ${(\frac{x}{z})}^{2} e^{{(\frac{z}{x})}^{2}} = \frac{e^{{(\frac{z}{x})}^{2}}}{{(\frac{z}{x})}^{2}} ⩾ e$ ${(\frac{y}{x})}^{2} e^{{(\frac{x}{y})}^{2}} = \frac{e^{{(\frac{x}{y})}^{2}}}{{(\frac{x}{y})}^{2}} ⩾ e$ ${(\frac{z}{y})}^{2} e^{{(\frac{y}{z})}^{2}} = \frac{e^{{(\frac{y}{z})}^{2}}}{{(\frac{y}{z})}^{2}} ⩾ e$ $\Rightarrow {(\frac{x}{z})}^{2} e^{{(\frac{z}{x})}^{2}} + {(\frac{y}{x})}^{2} e^{{(\frac{x}{y})}^{2}} + {(\frac{z}{y})}^{2} e^{{(\frac{y}{z})}^{2}} ⩾ e + e + e = 3 e$
Commented bymathdanisur last updated on 14/Jun/21

$c o o l S i r t h a n k y o u$
	Answered by SEIJacob last updated on 14/Jun/21


	Answered by mindispower last updated on 14/Jun/21

	$A M - H M$ $\Rightarrow$ $⩾ 3 \sqrt[3]{(\frac{x^{2} . y^{2} . z^{2}}{y^{2} . z^{2} . x^{2}}) e^{\frac{x^{2}}{y^{2}} + \frac{y^{2}}{z^{2}} + \frac{z^{2}}{x^{2}}}} = 3 e^{\frac{1}{3} (\frac{x^{2}}{y^{2}} + \frac{y^{2}}{z^{2}} + \frac{z^{2}}{x^{2}})}$ $A M - G M$ $\frac{x^{2}}{y^{2}} + \frac{z^{2}}{x^{2}} + \frac{y^{2}}{z^{2}} ⩾ 3 (\frac{x^{2} . y^{2} . z^{2}}{y^{2} . z^{2} . x^{2}}) = 3$ $\Leftrightarrow⩾ 3 e$
	Commented bymathdanisur last updated on 14/Jun/21

	$t h a n k s S i r c o o l$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum