Question

cho a,b,c,d>0, a+b+c+d=4

tìm gtnn: S=1/(a^2+1)+1/(b^2+1)+1/(c^2+1)+1/(d^2+1)

Answer 1

\(S=\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}+\dfrac{1}{d^2+1}\)

\(\dfrac{1}{a^2+1}=1-\dfrac{a^2}{a^2+1}\ge1-\dfrac{a^2}{2a}=1-\dfrac{a}{2}\)

\(tương\) \(tự\) \(với:\dfrac{1}{b^2+1};\dfrac{1}{c^2+1};\dfrac{1}{d^2+1}\)

\(\Rightarrow S\ge1-\dfrac{a}{2}+1-\dfrac{b}{2}+1-\dfrac{c}{2}+1-\dfrac{d}{2}=4-\left(\dfrac{a+b+c+d}{2}\right)=4-\dfrac{4}{2}=2\)

\(\Rightarrow min_S=2\Leftrightarrow a=b=c=d=1\)