十万火急 柯西不等式习题 d 求证1/(a-b)+1/(b-c）+1/(c-a)≥9/(a-d)0且

十万火急 柯西不等式习题 d 求证1/(a-b)+1/(b-c）+1/(c-a)≥9/(a-d)0且

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)^2a/(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

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