Question

Giúp mk vs!!

Tìm giá trị nhỏ nhất của biểu thức:

P=\(x^2-x\sqrt{y}+x+y-\sqrt{y}+1\)

Answer 1

\(P=x^2-x\sqrt{y}+x+y-\sqrt{y}+1\)

\(\Leftrightarrow2P=2x^2-2x\sqrt{y}+2x+2y-2\sqrt{y}+2\)

\(=\left[\left(x^2-2x\sqrt{y}+y\right)+\frac{4}{3}.\left(x-\sqrt{y}\right)+\frac{4}{9}\right]+\left(x^2+\frac{2}{3}x+\frac{1}{9}\right)+\left(y-\frac{2}{3}.\sqrt{y}+\frac{1}{9}\right)+\frac{4}{3}\)

\(=\left(x-\sqrt{y}+\frac{2}{3}\right)^2+\left(x+\frac{1}{3}\right)^2+\left(\sqrt{y}-\frac{1}{3}\right)^2+\frac{4}{3}\ge\frac{4}{3}\)

\(\Rightarrow P\ge\frac{2}{3}\)