Cho các số thực dương x,y,z thỏa mãn \(xy+yz+zx\ge x+y+z\).CMR:
\(\dfrac{x^2}{\sqrt{x^3+8}}+\dfrac{y^2}{\sqrt{y^3+8}}+\dfrac{z^2}{\sqrt{x^3+8}}\ge1\)
DƯƠNG PHAN KHÁNH DƯƠNG,Nguyễn Việt Lâm Giúp với !
\(\frac{\left(x+y+z\right)^2}{3}\ge xy+xz+yz\ge x+y+z\Rightarrow\left(x+y+z\right)^2\ge3\left(x+y+z\right)\)
\(\Rightarrow x+y+z\ge3\Rightarrow xy+xz+yz\ge3\)
\(\sum\frac{x^2}{\sqrt{x^3+8}}=\sum\frac{x^2}{\sqrt{\left(x+2\right)\left(x^2-2x+4\right)}}\ge\sum\frac{2x^2}{x^2-x+6}\ge\frac{2\left(x+y+z\right)^2}{x^2+y^2+z^2-\left(x+y+z\right)+18}\)
\(\Rightarrow VT\ge\frac{2\left(x+y+z\right)^2}{x^2+y^2+z^2+2\left(xy+xz+yz\right)-\left(x+y+z\right)+12}\ge\frac{2\left(x+y+z\right)^2}{\left(x+y+z\right)^2-\left(x+y+z\right)+12}\)
Đặt \(x+y+z=a\ge3\Rightarrow VT\ge\frac{2a^2}{a^2-a+12}=1+\frac{a^2+a-12}{a^2-a+12}=1+\frac{\left(a-3\right)\left(a+4\right)}{\left(a-\frac{1}{2}\right)^2+\frac{47}{4}}\)
Do \(a\ge3\Rightarrow\frac{\left(a-3\right)\left(a+4\right)}{\left(a-\frac{1}{2}\right)^2+\frac{47}{4}}\ge0\Rightarrow VT\ge1\) (đpcm)
Dấu "=" xảy ra khi \(x=y=z=1\)
