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Question Number 195288 by CrispyXYZ last updated on 29/Jul/23
x, y, z∈R_+ , P = (x/(x + y)) + (y/(y + z)) + (z/(z + x)), Q = (y/(x + y)) + (z/(y + z)) + (x/(z + x)), Q = (z/(x + y)) + (x/(y + z)) + (y/(z + x)). f(x, y, z)=max{P, Q, R}, find f_(min) .
$${x},\:{y},\:{z}\in\mathbb{R}_{+} , \\ $$$${P}\:=\:\frac{{x}}{{x}\:+\:{y}}\:+\:\frac{{y}}{{y}\:+\:{z}}\:+\:\frac{{z}}{{z}\:+\:{x}}, \\ $$$${Q}\:=\:\frac{{y}}{{x}\:+\:{y}}\:+\:\frac{{z}}{{y}\:+\:{z}}\:+\:\frac{{x}}{{z}\:+\:{x}}, \\ $$$${Q}\:=\:\frac{{z}}{{x}\:+\:{y}}\:+\:\frac{{x}}{{y}\:+\:{z}}\:+\:\frac{{y}}{{z}\:+\:{x}}. \\ $$$${f}\left({x},\:{y},\:{z}\right)=\mathrm{max}\left\{{P},\:{Q},\:{R}\right\},\:\mathrm{find}\:{f}_{\mathrm{min}} . \\ $$
Commented by Frix last updated on 29/Jul/23
(3/2)
$$\frac{\mathrm{3}}{\mathrm{2}} \\ $$

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