设a,b,c属于R正且 a+b+c=1,求(a+1/a)²+(b+1/b)²+(c+1/c)²的最
问题描述:
设a,b,c属于R正且 a+b+c=1,求(a+1/a)²+(b+1/b)²+(c+1/c)²的最
设a,b,c属于R正且
a+b+c=1,求(a+1/a)²+(b+1/b)²+(c+1/c)²的最小值
答
用柯西不等式(1+1+1)[(a+1/a)^2+(b+1/b)^2+(c+1/c)^2]>=(a+1/a+b+1/b+c+1/c)^2=(1+1/a+1/b+1/c)^2(a+b+c)(1/a+1/b+1/c)>=(1+1+1)^2=91/a+1/b+1/c>=9(1+1/a+1/b+1/c)^2>=(1+9)^2=100(1+1+1)[(a+1/a)^2+(b+1/b)^2+(c+1...