Áp dụng bđt svacxo: \(\frac{x_1^2}{y_1}+\frac{x_2^2}{y_2}\ge\frac{\left(x_1+x_2\right)^2}{y_1+y_2}\) (1)
CM bđt đúng: Từ (1) <=> \(\left(\frac{x_1^2}{y_1}+\frac{x_2^2}{y_2}\right)\left(y_1+y_2\right)\ge\left(x_1+x_2\right)^2\)
<=> \(x_1^2+\frac{x_1^2.y_2}{y_1}+\frac{x_2^2.y_1}{y_2}+x_2^2\ge x_1^2+2x_1x_2+x_2^2\)
<=> \(\frac{x_1^2y_2^2-2x_1x_2y_1y_2+x_2^2y_1^2}{y_1.y_2}\ge0\)
<=> \(\frac{\left(x_1y_2-x_2y_1\right)^2}{y_1y_2}\ge0\)(luôn đúng với mọi y1; y2 > 0)
Khi đó: F = \(\left(1+\frac{1}{a}\right)^2+\left(1+\frac{1}{b}\right)^2\ge\frac{\left(1+\frac{1}{a}+1+\frac{1}{b}\right)^2}{2}\ge\frac{\left(2+\frac{4}{a+b}\right)^2}{2}=\frac{\left(2+4\right)^2}{2}=18\)
Dấu "=" xảy ra <=> \(\hept{\begin{cases}1+\frac{1}{a}=1+\frac{1}{b}\\\frac{1}{a}=\frac{1}{b}\\a+b=1\end{cases}}\) <=> a = b = 1/2
Vậy MinF = 18 khi a = b = 1/2
