`#tnvt`
`M=\frac{x+3}{\sqrt{x}-1}(x>=0,x\ne1)`
`=\frac{x-1+4}{\sqrt{x}-1}`
`=\frac{(\sqrt{x}-1)(\sqrt{x}+1)+4}{\sqrt{x}-1}`
`=\sqrt{x}+1+\frac{4}{\sqrt{x}-1}`
`=\sqrt{x}-1+\frac{4}{\sqrt{x}-1}+2`
Áp dụng BĐT Cô-si, ta có:
`(\sqrt{x}-1)+\frac{4}{\sqrt{x}-1}>=2\sqrt{(\sqrt{x}-1).\frac{4}{\sqrt{x}-1}}=2\sqrt{4}=4`
`=>\sqrt{x}-1+\frac{4}{\sqrt{x}-1}+2>=4+2=6`
Dấu `=` xảy ra khi `\sqrt{x}-1=\frac{4}{\sqrt{x}-1}`
`<=>(\sqrt{x}-1)^2=4`
`<=>\sqrt{x}-1=2(\sqrt{x}-1>=-1)`
`<=>\sqrt{x}=3`
`<=>x=9(tm)`
Vậy `GTNNNN_M=6` khi `x=9`
