Question

Answer 1

-Áp dụng BĐT Caushy ta có: 

\(4\le\left(x+1\right)\left(y+1\right)\le\dfrac{\left[\left(x+1\right)+\left(y+1\right)\right]^2}{4}\)

\(\Rightarrow4\le\dfrac{\left[\left(x+1\right)+\left(y+1\right)\right]^2}{4}\)

\(\Leftrightarrow\left(x+y+2\right)^2\ge16\)

\(\Rightarrow x+y+2\ge4\) (do x,y>0)

\(\Leftrightarrow x+y\ge2\)

-Dấu "=" xảy ra \(\Leftrightarrow x=y=1\)

-Áp dụng BĐT Schwarz ta có:

\(P=\dfrac{x^2}{y}+\dfrac{y^2}{x}\ge\dfrac{\left(x+y\right)^2}{x+y}=x+y\ge2\)

-Dấu "=" xảy ra \(\Leftrightarrow x=y=1\)