[Toán 9]Bất đẳng thức

C

congchuaanhsang

Áp dụng bđt Cauchy cho 2 số dương ta có

$4a^2+b^2$\geq$2\sqrt{4a^2.b^2}$=4ab

\Leftrightarrow$4a^2+3b^2$\geq$4ab+2b^2$ \Leftrightarrow $a^2+3(a^2+b^2)$\geq$4ab+2b^2$

\Leftrightarrow$a^2+18$\geq$4ab+2b^2$ \Leftrightarrow $3a^2+18$\geq$2a^2+4ab+2b^2$

\Leftrightarrow$\sqrt{3(a^2+6)}$\geq$\sqrt{2(a^2+2ab+b^2)}$

\Leftrightarrow$\sqrt{3(a^2+6)}$\geq$\sqrt{2}(a+b)$

Dấu "=" xảy ra \Leftrightarrow a=$\dfrac{\sqrt{30}}{6}$ ; b=$\dfrac{2\sqrt{30}}{6}$
 
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