Question

设a,b,c∈R_,则a/(b+c)+b/(c+a)+c/(a+b)的最小值为

Answer 1

a/(b+c)+b/(c+a)+c/(a+b)=c/(a+b)+1+a/(b+c)+1+b/(c+a)+1-3

=(a+b+c)/(a+b)+(a+b+c)/(b+c)+(a+b+c)/(c+a)-3

=0.5*(a+b+b+c+c+a)*[1/(a+b)+1/(b+c)+1/(c+a)]-3

利用三元基本不等式

≥0.5*{3*[(a+b)(b+c)(c+a)]^1/3}*{3*[1/(a+b)*1/(b+c)*1/(c+a)]^1/3}-3

=0.5*3*3-3=3/2,

a/(b+c)+b/(c+a)+c/(a+b)的最小值为3/2.