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### springboot | ||
- mvn spring-boot:run | ||
- mvn package 打包后用java -jar来运行 | ||
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## 配置 | ||
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# 高中数学 | ||
> | ||
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<details> | ||
<summary>厦门2024强基</summary> | ||
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$$ | ||
\text{已知}x,y \gt 0, (x + y + xy)(x + y - xy) = xy, \text{求}x + y + xy \text{和}x + y - xy \text{的最小值}. \newline | ||
\text{解读:记}A = x + y, B = xy, \text{则有} \newline | ||
\begin{cases} | ||
x,y \gt 0, A^2 \ge 0 \to A^2 - 4B \ge 0 \to A^2 \ge 4B, & \text{(1)} \newline | ||
(A + B)(A - B)=B \to A^2 - B^2 = B \to A^2 = B^2 + B, & \text{(2)} \newline | ||
\end{cases} \newline | ||
\text{联立解得}B \ge 3 (\text{等号成立仅且当}x=y=\sqrt{3}) \newline | ||
x + y + xy = A + B \ge 2\sqrt{B} + B \ge 2\sqrt{3} + 3 \newline | ||
x + y - xy = A - B = \sqrt{B^2 + B} - B = \frac{B^2 + B - B^2}{\sqrt{B^2 + B} - B} = \frac{1}{ \sqrt{1 + \frac{1}{B}} + 1} | ||
\ge = \frac{1}{ \sqrt{1 + \frac{1}{3}} + 1} = 2\sqrt{3} - 3. | ||
$$ | ||
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式子(1)的推理处理后面的就容易了,一步步地推到就可以得到。算术平均数总是大于或等于几何平均数,即算术平均数-几何平均数不等式(AM-GM不等式) | ||
这个性质在数学与物理的许多领域都有应用 | ||
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</details> |
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