Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 190546 by cortano12 last updated on 05/Apr/23

$$\mathrm{Given}\mathrm{x},\mathrm{y},\mathrm{z}>0\mathrm{and}$$ $${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2+\mathrm{x}+2\mathrm{y}+3\mathrm{z}=23$$ $$\mathrm{find}\mathrm{maximum}\mathrm{of}\mathrm{x}+\mathrm{y}+\mathrm{z}.$$

Answered by mr W last updated on 05/Apr/23

$${(x+\frac{1}{2})}^2+{(y+1)}^2+{(z+\frac{3}{2})}^2=\frac{53}{2}$$ $$sayx+y+z=s$$ $$\sqrt{\frac{53}{2}}=\frac{\mid -\frac{1}{2}-1-\frac{3}{2}-s\mid }{\sqrt{3}}$$ $$s_{max}=-3+\sqrt{\frac{159}{2}}✓$$ $$s_{min}=-3-\sqrt{\frac{159}{2}}$$

Commented bycortano12 last updated on 05/Apr/23

$$\mathrm{by}\mathrm{Cauhcy}?$$

Commented bymr W last updated on 06/Apr/23

$$no.$$ $$planex+y+z=smusttangentthe$$ $$sphere{(x+\frac{1}{2})}^2+{(y+1)}^2+{(z+\frac{3}{2})}^2=\frac{53}{2}$$ $$\Rightarrow distancefromcenterofsphere$$ $$(-\frac{1}{2},-1,-\frac{3}{2})totheplaneisthe$$ $$radius\sqrt{\frac{53}{2}}.$$

Commented bycortano12 last updated on 07/Apr/23

$$\mathrm{ooyes}.\mathrm{thank}\mathrm{you}$$

Answered by mr W last updated on 07/Apr/23

$$MethodII$$ $$F=x+y+z-\frac{1}{\lambda }(x^2+y^2+z^2+x+2y+3z-23)$$ $$\frac{\partial F}{\partial x}=1-\frac{1}{\lambda }(2x+1)=0\Rightarrow x=\frac{1}{2}(\lambda -1)$$ $$\frac{\partial F}{\partial y}=1-\frac{1}{\lambda }(2y+2)=0\Rightarrow y=\frac{1}{2}(\lambda -2)$$ $$\frac{\partial F}{\partial z}=1-\frac{1}{\lambda }(2z+3)=0\Rightarrow z=\frac{1}{2}(\lambda -3)$$ $$\frac{{(\lambda -1)}^2}{4}+\frac{(\lambda -1)}{2}+\frac{{(\lambda -2)}^2}{4}+\frac{2(\lambda -2)}{2}+\frac{{(\lambda -3)}^2}{4}+\frac{3(\lambda -3)}{2}=23$$ $$3{\lambda }^2=106$$ $$\Rightarrow \lambda =\pm \sqrt{\frac{106}{3}}$$ $$x+y+z=-3+\frac{3\lambda }{2}=-3\pm \sqrt{\frac{159}{2}}$$ $${(x+y+z)}_{max}=-3+\sqrt{\frac{159}{2}}$$ $${(x+y+z)}_{min}=-3-\sqrt{\frac{159}{2}}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com