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# System prepended metadata

title: 柯西
tags: [數學問題]

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###### tags: `數學問題`
$\begin{align}\end{align}$
# 柯西

$a，b，c，x，y，z\in\mathbb{R}$，$a，b，c>0$，$a+b+c=1$，求證：

$$(ax+by+cz)^4\leq ax^4+by^4+cz^4$$

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$(a+b+c)^3(ax^4+by^4+cz^4)$
$=[(a^{\frac{1}{4}})^4+(b^{\frac{1}{4}})^4+(c^{\frac{1}{4}})^4]^3[(a^{\frac{1}{4}}x)^4+(b^{\frac{1}{4}}y)^4+(c^{\frac{1}{4}}z)^4]$
$\geq (ax+by+cz)^4$

