Question

1) Cho 2 số dương a,b. CMR: a+b1
2) Tìm GTNN:
M=\(\frac{1}{ab}+\frac{1}{a^2+ab}+\frac{1}{b^2+ab}+\frac{1}{a^2+b^2}\)

Answer 1

\(M=\frac{1}{ab}+\frac{1}{a^2+ab}+\frac{1}{b^2+ab}+\frac{1}{a^2+b^2}\)

\(=\left(\frac{1}{2ab}+\frac{1}{a^2+b^2}\right)+\left(\frac{1}{a^2+ab}+\frac{1}{b^2+ab}\right)+\frac{1}{2ab}\)

\(\ge\frac{\left(1+1\right)^2}{a^2+2ab+b^2}+\frac{\left(1+1\right)^2}{a^2+ab+b^2+ab}+\frac{2}{\left(a+b\right)^2}\)

\(=\frac{4}{\left(a+b\right)^2}+\frac{4}{\left(a+b\right)^2}+\frac{2}{\left(a+b\right)^2}\)

\(\ge\frac{4}{1}+\frac{4}{1}+\frac{2}{1}=10\)

Dấu = xảy ra khi a = b = \(\frac{1}{2}\)