Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 144917 by mathdanisur last updated on 30/Jun/21

if  x;y>0  then:  10 ∙ (√((x^2 +y^2 )/2)) + ((8xy)/(x+y)) ≥ 7x+7y

$${if}\:\:{x};{y}>\mathrm{0}\:\:{then}: \\ $$ $$\mathrm{10}\:\centerdot\:\sqrt{\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{\mathrm{2}}}\:+\:\frac{\mathrm{8}{xy}}{{x}+{y}}\:\geqslant\:\mathrm{7}{x}+\mathrm{7}{y} \\ $$

Answered by justtry last updated on 30/Jun/21

remember :  QM≥AM  (√((x^2 +y^2 )/2))≥((x+y)/2).....(i)  AM≥HM  ((x+y)/2)≥(2/((1/x) + (1/y))) =((2xy)/(x+y))   ⇔x+y≥((4xy)/(x+y))  ⇔2(x+y)≥((8xy)/(x+y)).....(ii)  10.(√((x^2 +y^2 )/2)) +((8xy)/(x+y)) ≥ 10.((x+y)/2) + 2x+2y                                           ≥5x+5y + 2x+2y                                          ≥7x+7y  10(√((x^2 +y^2 )/2)) + ((8xy)/(x+y)) ≥7x+7y

$${remember}\:: \\ $$ $${QM}\geqslant{AM} \\ $$ $$\sqrt{\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{\mathrm{2}}}\geqslant\frac{{x}+{y}}{\mathrm{2}}.....\left({i}\right) \\ $$ $${AM}\geqslant{HM} \\ $$ $$\frac{{x}+{y}}{\mathrm{2}}\geqslant\frac{\mathrm{2}}{\frac{\mathrm{1}}{{x}}\:+\:\frac{\mathrm{1}}{{y}}}\:=\frac{\mathrm{2}{xy}}{{x}+{y}}\: \\ $$ $$\Leftrightarrow{x}+{y}\geqslant\frac{\mathrm{4}{xy}}{{x}+{y}} \\ $$ $$\Leftrightarrow\mathrm{2}\left({x}+{y}\right)\geqslant\frac{\mathrm{8}{xy}}{{x}+{y}}.....\left({ii}\right) \\ $$ $$\mathrm{10}.\sqrt{\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{\mathrm{2}}}\:+\frac{\mathrm{8}{xy}}{{x}+{y}}\:\geqslant\:\mathrm{10}.\frac{{x}+{y}}{\mathrm{2}}\:+\:\mathrm{2}{x}+\mathrm{2}{y} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{5}{x}+\mathrm{5}{y}\:+\:\mathrm{2}{x}+\mathrm{2}{y} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\mathrm{7}{x}+\mathrm{7}{y} \\ $$ $$\mathrm{10}\sqrt{\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{\mathrm{2}}}\:+\:\frac{\mathrm{8}{xy}}{{x}+{y}}\:\geqslant\mathrm{7}{x}+\mathrm{7}{y} \\ $$ $$ \\ $$

Commented bymathdanisur last updated on 30/Jun/21

cool Sir, thank you

$${cool}\:{Sir},\:{thank}\:{you} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com