十万火急 柯西不等式习题 1.已知a>b>c>d 求证1/(a-b)+1/(b-c)+1/(c-a)≥9/(a-d)2.已知a,b,c>0且满足a+b+c=1 求证a3+b3+c3≥(a2+b2+c2)/33.若a,b,c>0,证明a/(b+2c)+b/(c+2a)+c/(a+2b)≥1注.字母后的数字为上标
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![十万火急 柯西不等式习题 1.已知a>b>c>d 求证1/(a-b)+1/(b-c)+1/(c-a)≥9/(a-d)2.已知a,b,c>0且满足a+b+c=1 求证a3+b3+c3≥(a2+b2+c2)/33.若a,b,c>0,证明a/(b+2c)+b/(c+2a)+c/(a+2b)≥1注.字母后的数字为上标](/uploads/image/z/3980468-20-8.jpg?t=%E5%8D%81%E4%B8%87%E7%81%AB%E6%80%A5+%E6%9F%AF%E8%A5%BF%E4%B8%8D%E7%AD%89%E5%BC%8F%E4%B9%A0%E9%A2%98+1.%E5%B7%B2%E7%9F%A5a%3Eb%3Ec%3Ed+%E6%B1%82%E8%AF%811%2F%28a-b%29%2B1%2F%28b-c%EF%BC%89%2B1%2F%28c-a%29%E2%89%A59%2F%28a-d%292.%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%3E0%E4%B8%94%E6%BB%A1%E8%B6%B3a%2Bb%2Bc%3D1+%E6%B1%82%E8%AF%81a3%2Bb3%2Bc3%E2%89%A5%28a2%2Bb2%2Bc2%29%2F33.%E8%8B%A5a%2Cb%2Cc%3E0%2C%E8%AF%81%E6%98%8Ea%2F%EF%BC%88b%2B2c%29%2Bb%2F%28c%2B2a%29%2Bc%2F%28a%2B2b%29%E2%89%A51%E6%B3%A8.%E5%AD%97%E6%AF%8D%E5%90%8E%E7%9A%84%E6%95%B0%E5%AD%97%E4%B8%BA%E4%B8%8A%E6%A0%87)
十万火急 柯西不等式习题 1.已知a>b>c>d 求证1/(a-b)+1/(b-c)+1/(c-a)≥9/(a-d)2.已知a,b,c>0且满足a+b+c=1 求证a3+b3+c3≥(a2+b2+c2)/33.若a,b,c>0,证明a/(b+2c)+b/(c+2a)+c/(a+2b)≥1注.字母后的数字为上标
十万火急 柯西不等式习题
1.已知a>b>c>d 求证1/(a-b)+1/(b-c)+1/(c-a)≥9/(a-d)
2.已知a,b,c>0且满足a+b+c=1 求证a3+b3+c3≥(a2+b2+c2)/3
3.若a,b,c>0,证明a/(b+2c)+b/(c+2a)+c/(a+2b)≥1
注.字母后的数字为上标
十万火急 柯西不等式习题 1.已知a>b>c>d 求证1/(a-b)+1/(b-c)+1/(c-a)≥9/(a-d)2.已知a,b,c>0且满足a+b+c=1 求证a3+b3+c3≥(a2+b2+c2)/33.若a,b,c>0,证明a/(b+2c)+b/(c+2a)+c/(a+2b)≥1注.字母后的数字为上标
只需证明 (a-d) [1/(a-b)+1/(b-c)+1/(c-d)] >= 9
[1/(a-b) +1/(b-c) +1/(c-d)](a-d)
=[1/(a-b) +1/(b-c) +1/(c-d)](a-b+b-c+c-d)
= [1 + (b-c)/(a-b) + (c-d)/(a-b)] + [1 + (a-b)/(b-c) + (c-d)/(b-c)] + [1 + [(a-b)/(c-d) + (b-c)/(a-d)]
= 3 + [(b-c)/(a-b) + (a-b)/(b-c)] + [(c-d)/(a-b) + (a-b)/(c-d)] + [(c-d)/(b-c) + (b-c)/(c-d)]
≥ 3 + 2 + 2 + 2
= 9
所以 :1/(a-b)+1/(b-c)+1/(c-d)>=9/(a-d)
3.若a,b,c>0,证明a/(b+2c)+b/(c+2a)+c/(a+2b)≥1
利用Cauchy-Schwarz不等式做
[a/(b+2c)+b/(c+2a)+c/(a+2b)]*(3ab+3bc+3ac)
= [a/(b+2c)+b/(c+2a)+c/(a+2b)]*[a(b+2c)+b(c+2a)+c(a+2b)]
≥(a+b+c)^2
a/(b+2c)+b/(c+2a)+c/(a+2b)≥(a+b+c)^2/(3ab+3bc+3ac)
因为 (a+b+c)^2 ≥ 3ab+3bc+3ac 所以
a/(b+2c)+b/(c+2a)+c/(a+2b)≥1,等号当且仅当 a=b=c时成立