question about optimization

Optimization
The level of satisfaction an employe has on its job is determined by many factors. Obviously, the salary plays an important role, but also the fact that a work is "clean"(relaxed) or "dirty"(stressful) are also important. Suppose that for a specific employe the level of satisfaction can be expressed by the following utility function: u = √w − 2x, where w
is the wage per hour and the parameter x ∈ [0, 1] denotes the level of "pollutionöf the job: x = 0 corresponds to a "clean"job and x = 1 a "dirty"job. "Dirty"jobs are usually paid better than "clean"jobs. The relation between w and x is given by w = 2+6x.
a) Is this employe more happier in a complete "clean"job or a complete "dirty"job?
b) Determine the level of dirtiness x this employe likes most and determine the corresponding
wage.

The utility function becomes

u=√(2+6x)-2x

Now, take the first derivative

((du)/(dx))=(3/(√(2+6x)))-2.

Set ((du)/(dx))=0, we obtain

(3/(√(2+6x)))-2=0,

which produces

x=(1/(24)).

Therefore, the answer to (a) is negative.
He like x=1/24 the most and wage is

w=2+6/24=9/4.

complete clean: x=0
wage w= 2 + 6(0) = 2
satisfaction u = √2 -2(0) = √2 ≈1.4142

complete dirty: x=1
wage w=2 + 6(1) = 8
satisfaction u = √8 -2(1) ≈2.8284 -2 ≈0.8284

a) Is this employe more happier in a complete "clean"job or a complete "dirty"job?
Since utility of clean job 1.4142 > utility of dirty job 0.8284, this empolyee is more happier in a complete clean job.