Đáp án:
$\begin{array}{l}
1)\\
{\left( {2x + 3y} \right)^2} = {\left( {\sqrt 2 .\sqrt 2 x + \sqrt 3 .\sqrt 3 y} \right)^2} \le \left( {2 + 3} \right).\left( {2{x^2} + 3{y^2}} \right)\\
\Rightarrow {4^2} \le 5.\left( {2{x^2} + 3{y^2}} \right)\\
\Rightarrow 2{x^2} + 3{y^2} \ge \frac{{16}}{5}\\
3)\\
{\left( {3x + 4y} \right)^2} \le \left( {{3^2} + {4^2}} \right)\left( {{x^2} + {y^2}} \right)\\
\Rightarrow {7^2} \le 25\left( {{x^2} + {y^2}} \right)\\
\Rightarrow {x^2} + {y^2} \ge \frac{{49}}{{25}}\\
7){\left( {x + 7y} \right)^2} \le \left( {1 + {7^2}} \right)\left( {{x^2} + {y^2}} \right)\\
\Rightarrow 100 \le 50\left( {{x^2} + {y^2}} \right)\\
\Rightarrow {x^2} + {y^2} \ge 2
\end{array}$
