回復 1# johncai 的帖子
用兩次科西不等式
\( \displaystyle \Bigg[\; \left( \frac{4}{x^3} \right)^2+\left( \frac{49}{y^3} \right)^2 \Bigg]\; (x^2+y^2)\ge \left( \frac{4}{x^2}+\frac{49}{y^2} \right)^2 \)
\( \displaystyle \Bigg[\; \left( \frac{2}{x} \right)^2+\left( \frac{7}{y} \right)^2 \Bigg]\; (x^2+y^2)\ge (2+7)^2 \)
故\( \displaystyle \Bigg[\; \left( \frac{4}{x^3} \right)^2+\left( \frac{49}{y^3} \right)^2 \Bigg]\; (x^2+y^2) \ge ((2+7)^2)^2=6561 \)
"="成立於\( \displaystyle \frac{\left(\displaystyle \frac{4}{x^3}\right)}{x}=\frac{\left(\displaystyle \frac{49}{y^3} \right)}{y} \)且\( \displaystyle \frac{\left(\displaystyle \frac{2}{x} \right)}{x}=\frac{\left(\displaystyle \frac{7}{y} \right)}{y} \)
\( \displaystyle \frac{x}{y}=\frac{\sqrt{14}}{7} \)
