Đặt ${x_1} + {x_2} + ... + {x_n}=m$Cho ${x_1}, {x_2}, ..., {x_n}\,\,(n \ge 2)$ là các số dương thỏa mãn:
${x_1} + {x_2} + ... + {x_n} \le k,\,\,(k\, \in {R^*}),\,b \ge 0,\,\,b{n^2} \ge a{k^2}$.
CMR: $a({x_1} + {x_2} + ... + {x_n}) + b\left( {\frac{1}{{{x_1}}} + \frac{1}{{{x_2}}} + ... + \frac{1}{{{x_n}}}} \right) \ge \frac{{b{n^2} + a{k^2}}}
{k}.$
Áp dụng CS, AM-GM, ta có :
$VT \ge am+\dfrac{bn^2}{m} =am+\dfrac{ak^2}{4m}+\dfrac{bn^2-ak^2}{m} \ge 2ak+\dfrac{bn^2-ak^2}{k} =\dfrac{bn^2+ak^2}{k}$