Question

已知a,b,c属于r+,证明a^2+b^2+c^2+(1÷a+1÷b+1÷c）^2≥6倍根号3,并确定a,b,c

Answer 1

a^2+b^2+c^2+(1/a+1/b+1/c)^2

=a^2+b^2+c^2+1/a^2+1/b^2+1/c^2+2/ab+2/bc+2/ca

>=a^2+b^2+c^2+3(1/ab+1/bc+1/ca)=(a^2+3/ab)+(b^2+3/bc)+(c^2+3/ca)

>=2√(3a/b)+2√(3b/c)+2√(3c/a)

>=6√3

a=b=c=四次根号3取等