Question Number 138004 by benjo_mathlover last updated on 09/Apr/21 | ||
$$ \\ $$ What is the minimum value of x+y+z, subject to the condition xyz = a³?\\n | ||
Answered by EDWIN88 last updated on 09/Apr/21 | ||
$${AM}−{GM} \\ $$ $$\:{for}\:{x},\:{y},\:{z}\:\geqslant\:\mathrm{0}\:\Rightarrow\:\frac{{x}+{y}+{z}}{\mathrm{3}}\:\geqslant\:\sqrt[{\mathrm{3}}]{{xyz}}\:;\:{x}+{y}+{z}\:\geqslant\:\mathrm{3}\:\sqrt[{\mathrm{3}}]{{a}^{\mathrm{3}} } \\ $$ $${minimum}\:{x}+{y}+{z}\:=\:\mathrm{3}{a} \\ $$ | ||
Answered by bobhans last updated on 09/Apr/21 | ||
$${f}\left({x},{y},{z},\lambda\right)={x}+{y}+{z}+\lambda\left({xyz}−{a}^{\mathrm{3}} \right) \\ $$ $$\frac{\partial{f}}{\partial{x}}\:=\:\mathrm{1}+\lambda{yz}=\mathrm{0} \\ $$ $$\frac{\partial{f}}{\partial{y}}=\mathrm{1}+\lambda{xz}\:=\:\mathrm{0} \\ $$ $$\frac{\partial{f}}{\partial{z}}=\mathrm{1}+\lambda{xy}=\mathrm{0} \\ $$ $$\Leftrightarrow\:−\frac{\mathrm{1}}{{yz}}=−\frac{\mathrm{1}}{{xz}}=−\frac{\mathrm{1}}{{xy}} \\ $$ $${we}\:{get}\:{xy}={xz}={yz}\:;\:{x}={y}={z} \\ $$ $${from}\:{condition}\:{xyz}\:=\:{a}^{\mathrm{3}} \\ $$ $${we}\:{get}\:{x}={y}={z}={a} \\ $$ $${minimum}\:{x}+{y}+{z}\:=\:\mathrm{3}{a} \\ $$ | ||
Commented bymr W last updated on 09/Apr/21 | ||
$${how}\:{to}\:{explain}\:{this}\:{example}: \\ $$ $${a}=\mathrm{1} \\ $$ $${x}=−\mathrm{10},\:{y}=−\frac{\mathrm{1}}{\mathrm{10}},\:{z}=\mathrm{1} \\ $$ $${xyz}=\left(−\mathrm{10}\right)\left(−\frac{\mathrm{1}}{\mathrm{10}}\right)\left(\mathrm{1}\right)=\mathrm{1}={a}^{\mathrm{3}} \\ $$ $${but}\:{x}+{y}+{z}=−\mathrm{10}−\frac{\mathrm{1}}{\mathrm{10}}+\mathrm{1}=−\mathrm{9}\frac{\mathrm{1}}{\mathrm{10}}<\mathrm{3}{a}=\mathrm{3} \\ $$ $${i}.{e}.\:\mathrm{3}{a}\:{is}\:{not}\:{minimum} \\ $$ | ||
Commented bymr W last updated on 09/Apr/21 | ||
$${i}\:{think}\:{x}+{y}+{z}\:{has}\:{no}\:{minimum}\:{or} \\ $$ $${maximum}\:{under}\:{the}\:{condition}\:{that} \\ $$ $${xyz}={a}^{\mathrm{3}} . \\ $$ | ||
Commented bymr W last updated on 09/Apr/21 | ||
$${i}\:{just}\:{wanted}\:{to}\:{show}\:{that}\:{x}={y}={z} \\ $$ $${doesn}'{t}\:{lead}\:{to}\:{minimum}\:{of}\:{x}+{y}+{z}. \\ $$ $${with}\:{x}=−\mathrm{10}\:{it}\:{is}\:{only}\:{an}\:{example}. \\ $$ $${i}\:{can}\:{also}\:{take}\:{an}\:{other}\:{example}: \\ $$ $${x}=\mathrm{1},\:{y}=−\mathrm{100},\:{z}=−\frac{\mathrm{1}}{\mathrm{100}}.\:{i}\:{fulfill} \\ $$ $${xyz}=\mathrm{1}={a}^{\mathrm{3}} \\ $$ $${but}\:{x}+{y}+{z}=−\mathrm{99}\frac{\mathrm{1}}{\mathrm{100}}\:{which}\:{is}\:{smaller} \\ $$ $${than}\:{the}\:{minimum}\:\mathrm{3}{a}=\mathrm{3}\:{you}\:{calculated}. \\ $$ | ||
Commented bybobhans last updated on 09/Apr/21 | ||
$${why}\:{you}\:{get}\:{x}=−\mathrm{10}?\:{from}\:\lambda\:{it} \\ $$ $${clear}\:{x}={y}={z} \\ $$ | ||