a,b,c>0,a+b+c=1.证明(a+1/a)^2+(b+1/b)^2+(c+1/c)^2>=100/3用柯西不等式解

a,b,c>0,a+b+c=1.证明(a+1/a)^2+(b+1/b)^2+(c+1/c)^2>=100/3用柯西不等式解
a,b,c>0,a+b+c=1.证明(a+1/a)^2+(b+1/b)^2+(c+1/c)^2>=100/3
用柯西不等式解

a,b,c>0,a+b+c=1.证明(a+1/a)^2+(b+1/b)^2+(c+1/c)^2>=100/3用柯西不等式解
先用一次柯西不等式：(a+b+c)(1/a+1/b+1/c)≥[√a*1/(√a)+√b*1/(√b)+√c*1/(√c)]^2=(1+1+1)^2,
则1/a+1/b+1/c≥9,
再用一次：[(a+1/a)^2+(b+1/b)^2+(c+1/c)^2](1+1+1)
≥(a+1/a+b+1/b+c+1/c)^2≥(1+9)^2,
3除过去,(a+1/a)^2+(b+1/b)^2+(c+1/c)^2≥100/3,得证.