Question

cho a>0,b>0 va a+b<=1. tim gtnn cua a^2+b^2+1/a^2+1/b^2

 

Answer 1

Ta co:\(1\ge a+b\ge2\sqrt{ab}\Rightarrow ab\le\frac{1}{4}\)

Dat \(P=a^2+b^2+\frac{1}{a^2}+\frac{1}{b^2}\)

\(=a^2+\frac{1}{16a^2}+b^2+\frac{1}{16b^2}+\frac{15}{16}\left(\frac{1}{a^2}+\frac{1}{b^2}\right)\)

\(=a^2+\frac{1}{16a^2}+b^2+\frac{1}{16b^2}+\frac{15}{16}.\frac{a^2+b^2}{a^2b^2}\ge\frac{1}{2}+\frac{1}{2}+\frac{15}{16}.\frac{2}{ab}\ge1+\frac{15}{16}.\frac{2}{\frac{1}{4}}=\frac{17}{2}\)

Dau '=' xay ra \(a=b=\frac{1}{2}\)

Vay \(P_{min}=\frac{17}{2}\)khi \(a=b=\frac{1}{2}\)