Question

Cho a^2 + b^2 - a*b = 4 Tìm Max . MIn M = a^2 + b^2

Answer 1

(*) tìm max 

\(a^2+b^2-ab=4\Leftrightarrow2\left(a^2+b^2-ab\right)=8\)

<=> \(\left(a-b\right)^2+a^2+b^2=8\)

<=> \(\left(a-b\right)^2=8-\left(a^2+b^2\right)\) . Vì \(\left(a-b\right)^2\ge0\) 

=> \(8-M\ge0\Leftrightarrow M\le8\) 

Vậy Max M = 8 khi x = y = 2 hoặc x = y= - 2 

(*) tìm Min

\(a^2+b^2-ab=4\Leftrightarrow2\left(a^2+b^2\right)=8+2ab\Leftrightarrow3\left(a+b\right)^2=8+\left(a+b\right)^2\)

Vì \(8+\left(a+b\right)^2\ge8\Leftrightarrow3M\ge8\Leftrightarrow M\ge\frac{8}{3}\) 

Vậy Min M = 8/3 khi x = -y = ....

Answer 2

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