if-x-y-z-gt-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 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 \dots

{(\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 by mr 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 by mathdanisur 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 by mr 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 by mathdanisur 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 by mathdanisur last updated on 14/Jun/21

t h a n k s S i r c o o l