Tìm GTNN của P: `\frac{x+8}{\sqrt{x}+1}`

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`\frac{x+8}{\sqrt{x}+1}(x>=0)`

`=\frac{x-1+9}{\sqrt{x}+1}`

`=\frac{(\sqrt{x}-1)(\sqrt{x}+1)+9}{\sqrt{x}+1}`

`=\sqrt{x}-1+\frac{9}{\sqrt{x}+1}`

`=\sqrt{x}+1+\frac{9}{\sqrt{x}+1}-2`

`\text{Sử dụng Bất Đẳng Thức Cauchy, ta có:}`

`(\sqrt{x}+1)+\frac{9}{\sqrt{x}+1}>=2\sqrt{(\sqrt{x}+1).\frac{9}{\sqrt{x}+1}}=2\sqrt{9}=6`

`=>\sqrt{x}+1+\frac{9}{\sqrt{x}+1}-2>=6-2=4`

Dấu "=" xảy ra khi `\sqrt{x}+1=\frac{9}{\sqrt{x}+1}`

`<=>(\sqrt{x}+1)^2=9`

Ta có: `\sqrt{x}>=0`

`=>\sqrt{x}+1>=1`

`=>\sqrt{x}+1=3`

`<=>\sqrt{x}=2`

`<=>x=4(\text{Thỏa mãn})`

Vậy `\text{Min}_P=4` khi `x=4`