已知a,b,c属于R,a^2+b^2+c^2=1.求证,|a+b+c|=(a+b+c)^2对满足题条件的实数a,b,c恒成立,求实数X的范围
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![已知a,b,c属于R,a^2+b^2+c^2=1.求证,|a+b+c|=(a+b+c)^2对满足题条件的实数a,b,c恒成立,求实数X的范围](/uploads/image/z/10146663-63-3.jpg?t=%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%E5%B1%9E%E4%BA%8ER%2Ca%5E2%2Bb%5E2%2Bc%5E2%3D1.%E6%B1%82%E8%AF%81%2C%7Ca%2Bb%2Bc%7C%3D%EF%BC%88a%2Bb%2Bc%29%5E2%E5%AF%B9%E6%BB%A1%E8%B6%B3%E9%A2%98%E6%9D%A1%E4%BB%B6%E7%9A%84%E5%AE%9E%E6%95%B0a%2Cb%2Cc%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82%E5%AE%9E%E6%95%B0X%E7%9A%84%E8%8C%83%E5%9B%B4)
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已知a,b,c属于R,a^2+b^2+c^2=1.求证,|a+b+c|=(a+b+c)^2对满足题条件的实数a,b,c恒成立,求实数X的范围
已知a,b,c属于R,a^2+b^2+c^2=1.求证,|a+b+c|=(a+b+c)^2对满足题条件的实数a,b,c恒成立,求实数X的范围
已知a,b,c属于R,a^2+b^2+c^2=1.求证,|a+b+c|=(a+b+c)^2对满足题条件的实数a,b,c恒成立,求实数X的范围
(1) (a+b+c)²=a²+b²+c²+2ab+2bc+2ca
≤a²+b²+c²+a²+b²+b²+c²+c²+a²
=3(a²+b²+c²)=3
所以 |a+b+c|≤√3
(2)|x-1|+|x+1|≥(a+b+c)²,
从而 |x-1|+|x+1|≥[(a+b+c)²]max
即 |x-1|+|x+1|≥3
故只需 |x+1+x-1|≥3
|x|≥3/2
x≥3/2或x≤-3/2.