Calculus help! Natural log functions

Let f be the function defined by f(x) = 2xln(2x)
a. What is the domain of f(x)? (I got (0,infinity))
b. Find the absolute minimum value of f.
c. What is the range of f?
d. Consider the family of functions defined by y = bxlnbx, where b is a nonzero constant. Show that the absolute minimum value of bxlnbx is the same for all nonzero values of b.

thanks!!

a.
2x > 0, x > 0

b.
f'(x) = 2ln(2x) + 2
0 = 2ln(2x) + 2
0 = ln(2x) + 1
ln(2x) = -1
2x = 1/e
x = 1/(2e)
f(1/(2e)) = (1/e)ln(1/e) = -1/e
So the minimum of f is -1/e.

c.
f ≥ -1/e

d.
f(x) = bxln(bx)
f'(x) = bln(bx) + b
0 = bln(bx) + b
0 = ln(bx) + 1
ln(bx) = -1
bx = 1/e
x = 1/(be)
f(1/(be)) = (1/e)ln(1/e) = -1/e
So, regardless of b, the absolute minimum of f(x) = bxln(bx) is -1/e.