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 157168 by mathocean1 last updated on 20/Oct/21

x , y and z are numbers.  Show that max(x, y)=((x+y+∣x−y∣)/2) and min(x,y)=((x+y−∣x−y∣)/2)  then find a formula for   max(x,y,z).

$${x}\:,\:{y}\:{and}\:{z}\:{are}\:{numbers}. \\ $$$${Show}\:{that}\:{max}\left({x},\:{y}\right)=\frac{{x}+{y}+\mid{x}−{y}\mid}{\mathrm{2}}\:{and}\:{min}\left({x},{y}\right)=\frac{{x}+{y}−\mid{x}−{y}\mid}{\mathrm{2}} \\ $$$${then}\:{find}\:{a}\:{formula}\:{for}\: \\ $$$${max}\left({x},{y},{z}\right). \\ $$

Answered by puissant last updated on 20/Oct/21

→max(x;y)=(1/2)max(2x;2y)  =(1/2)max(x−y+x+y ; y−x+x+y)  =(1/2)(x+y+max(x−y ; y−x)  =(1/2)(x+y+∣x−y∣) = ((x+y+∣x−y∣)/2)  →min(x;y)=(1/2)min(2x;2y)  =(1/2)min(x−y+x+y ; y−x+x+y)  =(1/2)(x+y+min(x−y ; y−x))  =(1/2)(x+y−∣x−y∣)=((x+y−∣x−y∣)/2)  → max(x ; y ; z)= max(max(x;y);z)    =((max(x;y)+z+∣max(x;y)−z∣)/2)  =((((x+y+∣x−y∣)/2)+z+∣((x+y+∣x−y∣)/2)−z∣)/2)..               ............Le puissant...........

$$\rightarrow{max}\left({x};{y}\right)=\frac{\mathrm{1}}{\mathrm{2}}{max}\left(\mathrm{2}{x};\mathrm{2}{y}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{max}\left({x}−{y}+{x}+{y}\:;\:{y}−{x}+{x}+{y}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}+{max}\left({x}−{y}\:;\:{y}−{x}\right)\right. \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}+\mid{x}−{y}\mid\right)\:=\:\frac{{x}+{y}+\mid{x}−{y}\mid}{\mathrm{2}} \\ $$$$\rightarrow{min}\left({x};{y}\right)=\frac{\mathrm{1}}{\mathrm{2}}{min}\left(\mathrm{2}{x};\mathrm{2}{y}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}{min}\left({x}−{y}+{x}+{y}\:;\:{y}−{x}+{x}+{y}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}+{min}\left({x}−{y}\:;\:{y}−{x}\right)\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left({x}+{y}−\mid{x}−{y}\mid\right)=\frac{{x}+{y}−\mid{x}−{y}\mid}{\mathrm{2}} \\ $$$$\rightarrow\:{max}\left({x}\:;\:{y}\:;\:{z}\right)=\:{max}\left({max}\left({x};{y}\right);{z}\right) \\ $$$$ \\ $$$$=\frac{{max}\left({x};{y}\right)+{z}+\mid{max}\left({x};{y}\right)−{z}\mid}{\mathrm{2}} \\ $$$$=\frac{\frac{{x}+{y}+\mid{x}−{y}\mid}{\mathrm{2}}+{z}+\mid\frac{{x}+{y}+\mid{x}−{y}\mid}{\mathrm{2}}−{z}\mid}{\mathrm{2}}.. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:............\mathscr{L}{e}\:{puissant}........... \\ $$

Commented by mathocean1 last updated on 22/Oct/21

thanks le puissant.

$${thanks}\:{le}\:{puissant}. \\ $$

Commented by puissant last updated on 24/Oct/21

de rien...

$${de}\:{rien}... \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com