Đáp án + Giải thích các bước giải:
`P=(x-2\sqrtx)/(\sqrtx+1)`
`=>P=(x+2\sqrtx+1+3-4\sqrtx-4)/(\sqrtx+1)`
`=>P=(x+2\sqrtx+1)/(\sqrtx+1)+3/(\sqrtx+1)-(4\sqrtx+4)/(\sqrtx+1)`
`=>P=(\sqrtx+1)^2/(\sqrtx+1)+3/(\sqrtx+1)-(4(\sqrtx+1))/(\sqrtx+1)`
`=>P=\sqrtx+1+3/(\sqrtx+1)-4`
Áp dụng BĐT Cosi ta có: `\sqrtx+1+3/(\sqrtx+1)\ge2\sqrt{(\sqrtx+1). 3/(\sqrtx+1)}=2\sqrt3`
`=>P=\sqrtx+1+3/(\sqrtx+1)-4\ge2\sqrt3-4`
Dấu `=` xảy ra khi: `\sqrtx+1=3/(\sqrtx+1)`
`=>\sqrtx+1-3/(\sqrtx+1)=0`
`=>(x+\sqrtx)/(\sqrtx+1)+(\sqrtx+1)/(\sqrtx+1)-3/(\sqrtx+1)=0`
`=>(x+\sqrtx+\sqrtx+1-3)/(\sqrtx+1)=0`
`=>(x+2\sqrtx-2)/(\sqrtx+1)=0`
`=>x+2\sqrtx-2=0`
`=>x+2\sqrtx+1=3`
`=>(\sqrtx+1)^2=3`
`=>\sqrtx+1=\pm\sqrt3`
`=>\sqrtx=\pm\sqrt3-1`
`=>x=3-2\sqrt3+1`
`=>x=4-2\sqrt3`
Vậy `P_(min)=2\sqrt3-4` khi `x=4-2\sqrt3`
