Question

Cho a,b > 0 và a + b = 1 Min A = ( 1 + 1/a)* ( 1+1/b)

Answer 1

\(A=\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\)

\(A=1+\frac{1}{b}+\frac{1}{a}+\frac{1}{ab}\)

\(A=1+\frac{a+b}{ab}+\frac{a+b}{ab}\)

\(A=1+\frac{2}{ab}\)

Ta có : 

\(\left(a-b\right)^2\ge0\)

\(\Rightarrow a^2+b^2\ge2ab\)

\(\Rightarrow\left(a+b\right)^2\ge4ab\)

\(\Rightarrow ab\le\frac{1}{4}\)

\(\Rightarrow A\ge1+\frac{2}{\frac{1}{4}}=9\)

" = " \(\Leftrightarrow a=b=0,5\)

Chúc bạn học tốt !!!