the exopnential and log funtion

Let f(x) = x^(1/2) - In x
d(f(x))/dx = 1/2/ x^(1/2) - 1/x
=[x^(1/2)-2]/[2x]
d(f(x))/dx =0 when x=4
and after verification, f(x) attains minimum at x=4
since f(4) = 2-ln2 >0 and f(x) >=f(4)
therefore x^(1/2) - In x >0 when x>0

thx