Áp dụng BĐT Bunyacovsky cho 2 bộ số ta có:
a) \(Y=\sqrt{1.\left(8+3x\right)}+\sqrt{\frac{1}{3}.\left(15-3x\right)}\le\sqrt{\left(1+\frac{1}{3}\right)\left(8+3x+15-3x\right)}\)
\(=\sqrt{\frac{92}{3}}\)
\(maxY=\sqrt{\frac{92}{3}}\Leftrightarrow x=\frac{37}{12}\)
b) \(Y=\sqrt{\frac{1}{3}.\left(21x-9\right)}+\sqrt{\frac{1}{7}.\left(28-21x\right)}\le\sqrt{\left(\frac{1}{3}+\frac{1}{7}\right)\left(21x-9+28-21\right)}=\sqrt{\frac{190}{21}}\)
\(maxY=\sqrt{\frac{190}{21}}\Leftrightarrow x=\frac{223}{210}\)
c) \(Y=\sqrt{\frac{1}{3}.\left(24x-15\right)}+\sqrt{\frac{1}{4}\left(40-24x\right)}\le\sqrt{\left(\frac{1}{3}+\frac{1}{4}\right)\left(24x-15+40-24x\right)}=\sqrt{\frac{175}{12}}\)
\(maxY=\sqrt{\frac{175}{12}}\Leftrightarrow x=\frac{11}{6}\)
Đính chính lại câu c: \(maxY=\sqrt{\frac{175}{12}}\Leftrightarrow x=\frac{205}{168}\)
