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 178818 by Shrinava last updated on 21/Oct/22

$$\mathrm{If}\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\in [1,2]\mathrm{then}\mathrm{prove}\mathrm{that}:$$$$(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e})(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}+\frac{1}{\mathrm{d}}+\frac{1}{\mathrm{e}})⩽28$$$$\mathrm{When}\mathrm{equality}\mathrm{holds}?$$

Answered by mr W last updated on 22/Oct/22

$$A=a+b+c+d+e$$$$B=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}$$$$\mathrm{\Phi }=A\times B$$$$wesee,whenwemakeAlarger,we$$$$makeBsmallerinthesametime.$$$$tomakeAlarger,weshouldmake$$$$asmanynumbersaspossibleequal$$$$to2.buttomakeBlarger,weshould$$$$makeasmanynumbersaspossible$$$$equalto1.$$$$wehave5numbers.{let}^{\prime }ssaynnumbers$$$$equal2and5-nnumbersequal1.$$$$A=n\times 2+(5-n)\times 1=n+5$$$$B=n\times \frac{1}{2}+(5-n)\times \frac{1}{1}=5-\frac{n}{2}$$$$\mathrm{\Phi }=(n+5)(5-\frac{n}{2})=\frac{50+5n-n^2}{2}$$$$=\frac{225}{8}-\frac{1}{2}{(n-\frac{5}{2})}^2$$$$sincenisinteger,n\ne \frac{5}{2}.$$$${\mathrm{\Phi }}_{max}isatn=2or3.$$$${\mathrm{\Phi }}_{max}=\frac{225}{8}-\frac{1}{2}\times {\left(\frac{1}{2}\right)}^2=\frac{224}{8}=28$$$$i.e.\mathrm{\Phi }⩽28.theequalityholdswhen$$$$2numbersequalto2and3numbers$$$$equalto1,orwhen3numbersequal$$$$to2and2umbersequalto1.$$

Commented by Tawa11 last updated on 22/Oct/22

$$\mathrm{Great}\mathrm{sir}$$

Commented by Shrinava last updated on 23/Oct/22

$$\mathrm{cool}\mathrm{dear}\mathrm{professor}\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com