`1.`
`A = (sqrt(x)+1)/(x+4sqrt(x)+4) : ( (x)/(x+2sqrt(x)) + x/(sqrt(x)+2) ) \ \ (x>0)`
`= ( sqrt(x)+1 )/((sqrt(x)+2)^2) : ( x/(sqrt(x).(sqrt(x)+2)) + x/(sqrt(x)+2) )`
`= (sqrt(x)+1)/((sqrt(x)+2)^2) : ( x + x . sqrt(x) )/(sqrt(x).(sqrt(x)+2))`
`= (sqrt(x)+1)/((sqrt(x)+2)^2) : (sqrt(x).(sqrt(x)+x))/(sqrt(x).(sqrt(x)+2))`
`= (sqrt(x)+1)/((sqrt(x)+2)^2) : (sqrt(x)+x)/(sqrt(x)+2)`
`= (sqrt(x)+1)/((sqrt(x)+2)^2) . ( sqrt(x)+2)/(sqrt(x)+x)`
`= (sqrt(x)+1)/((sqrt(x)+2)(sqrt(x)+x))`
`= (sqrt(x)+1)/((sqrt(x)+2).sqrt(x).(1+sqrt(x)))`
`= 1/(sqrt(x).(sqrt(x)+2))`
`= 1/(x+2sqrt(x))`
`2.`
`A >= 1/(3sqrt(x))`
`<=> 1/(x+2sqrt(x)) >= 1/(3sqrt(x))`
`<=> 1/(x+2\sqrt{x})-1/(3\sqrt{x}) >=0`
`<=> (3\sqrt{x}-x-2\sqrt{x})/(3\sqrt{x}(x+2\sqrt{x}))>=0`
`<=> (\sqrt{x}-x)/(3x(\sqrt{x}+2))>=0`
Do $x>0; \sqrt{x}+2>0 \ \forall x$
`to sqrt(x)-x>=0`
`<=> x=1`
