已知 $a,b,cinmathbb R$，求证：$|a|+|b|+|c|+|a+b+c|geqslant |a+b|+|b+c|+|c+a|$
分析:不妨设$c=max{a,b,c},dfrac{a}{c}=x,dfrac{b}{c}=y$两边同除$|c|$后只需证明
$|x|+|y|+1+|x+y+1|ge|x+y|+|y+1|+|x+1|$
注意到恒等式$|x|+|y|+|z|=max{|x+y+z|,|x+y-z|,|x-y+z|,|x-y-z|}$,易得.
练习:
设实数$x,y,z$满足
egin{equation}
left{ egin{aligned}
|x+2y-3z|& le6\
|x-2y+3z|&le6\
|x-2y-3z|&le 6\
|x+2y+3z|&le6\
end{aligned}
ight.
end{equation}
则$|x|+|y|+|z|$的最大值为_____
答案:6
提示:注意到恒等式$|x|+|y|+|z|=max{|x+y+z|,|x+y-z|,|x-y+z|,|x-y-z|}$,易得.