Inequality
Procedure: Sugar water inequality
Inequality of arithmetic and geometric means
NCEE
Make Up Proportion
来源： 疑似杭四中
\( x, y, z \in R^+\) 已知 \( x + y + z = 1 \) 求 \( \sqrt{x y} + \sqrt{x z} - y - z \) 的最大值
Solution
\( \sqrt{x y} + \sqrt{x z} - y - z = \sqrt{\lambda x (\frac{1}{\lambda}) y} + \sqrt{\mu x (\frac{1}{\mu}) z} - y - z \leq \frac{\lambda + \mu}{2} x + (\frac{1}{2 \lambda} - 1) y + (\frac{1}{2 \mu} - 1) z \)
\( \frac{\lambda + \mu}{2} = \frac{1}{2 \lambda} - 1 = \frac{1}{2 \mu} - 1 \)
......
\(\frac{\sqrt{3}-1}{2}\)
Taylor series (Maclaurin series)
\(\displaystyle \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n\) \(\displaystyle \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n\)
Sometimes the first few terms of the series is used.