Đáp án+Giải thích các bước giải:
`P=(x^2-sqrtx)/(x+sqrtx+1)-(2x+sqrtx)/sqrtx+(2(x-1))/(sqrtx-1)(x>0,x ne 1)`
`P=(sqrtx(xsqrtx-1))/(x+sqrtx+1)-(sqrtx(2sqrtx+1))/sqrtx+(2(sqrtx-1)(sqrtx+1))/(sqrtx-1)`
`P=(sqrtx(sqrtx-1)(x+sqrtx+1))/(x+sqrtx+1)-(2sqrtx+1)+2(sqrtx+1)`
`P=sqrtx(sqrtx-1)-2sqrtx-1+2sqrtx+2`
`P=x-sqrtx+1`
`b)P=x-sqrtx+1/4+3/4`
`P=(sqrtx-1/2)^2+3/4>=3/4`
`=>P>=3/4`
Dấu "=" xảy ra khi `sqrtx=1/2<=>x=1/4.`
`c)Q=(2sqrtx)/P`
`Q=(2sqrtx)/(x-sqrtx+1)`
`x>0=>2sqrtx>0`
`x-sqrtx+1=(sqrtx-1/2)^2+3/4>=3/4>0`
`=>Q>0`
Mặt khác:
`Q=2/(sqrtx-1+1/sqrtx)`
Áp dụng cosi:
`sqrtx+1/sqrtx>=2`
`=>sqrtx-1+1/sqrtx>=1`
`=>Q<=2`
Dấu "=" xảy ra khi `sqrtx=1/sqrtx<=>x=1(KTMĐKXĐ)`
`=>Q<2`
Mà `Q in ZZ`
`=>Q=1`
`<=>2sqrtx=x-sqrtx+1`
`<=>x-3sqrtx+1=0`
`<=>(sqrtx-3/2)^2=5/4`
`<=>sqrtx-3/2=(+-sqrt5)/2`
`<=>sqrtx=(+-sqrt5+3)/2`
`<=>x=(+-sqrt5+3)^2/2`
