Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 194257 by tri26112004 last updated on 01/Jul/23

$$Knowx,y,z\in R^{+}suchthat:$$$$2x+4y+7z=2xyz$$$$FindMin(x+y+z)¿$$

Commented by Frix last updated on 01/Jul/23

$$x=3\wedge y=\frac{5}{2}\wedge z=2\Rightarrow \mathrm{answer}\mathrm{is}\frac{15}{2}$$

Commented by tri26112004 last updated on 01/Jul/23

$$Canyouexplainit¿$$

Answered by mr W last updated on 01/Jul/23

$$F=x+y+z+\lambda (2x+4y+7z-2xyz)$$$$\frac{\partial F}{\partial x}=1+\lambda (2-2yz)=0\Rightarrow \frac{1}{2\lambda }+1=yz$$$$\frac{\partial F}{\partial y}=1+\lambda (4-2xz)=0\Rightarrow \frac{1}{2\lambda }+2=xz$$$$\frac{\partial F}{\partial z}=1+\lambda (7-2xy)=0\Rightarrow \frac{1}{2\lambda }+\frac{7}{2}=xy$$$$let\xi =\frac{1}{2\lambda }$$$$yz=\xi +1$$$$xz=\xi +2$$$$xy=\xi +\frac{7}{2}$$$$\frac{2}{yz}+\frac{4}{xz}+\frac{7}{xy}=2$$$$\frac{2}{\xi +1}+\frac{4}{\xi +2}+\frac{7}{\xi +\frac{7}{2}}=2$$$$2{\xi }^3-25\xi -28=0$$$$(\xi -4)(2{\xi }^2+8\xi +7)=0$$$$\Rightarrow \xi =4>-1✓$$$$\Rightarrow \xi =\frac{-4\pm \sqrt{2}}{2}<-1\Rightarrow rejected$$$$yz=5...\left(i\right)$$$$xz=6...\left(ii\right)$$$$xy=\frac{15}{2}...\left(iii\right)$$$$\Rightarrow x=\sqrt{\frac{6\times \frac{15}{2}}{5}}=3$$$$\Rightarrow y=\sqrt{\frac{5\times \frac{15}{2}}{6}}=\frac{5}{2}$$$$\Rightarrow z=\sqrt{\frac{5\times 6}{\frac{15}{2}}}=2$$$$\Rightarrow {(x+y+z)}_{min}=3+\frac{5}{2}+2=\frac{15}{2}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com