5. A company makes a particular type of watch.
The annual profit made by the company from sales of these watches is modelled by the equation
$$P = 12 x - x ^ { \frac { 3 } { 2 } } - 120$$
where \(P\) is the annual profit measured in thousands of pounds and \(\pounds x\) is the selling price of the watch.
According to this model,
- find, using calculus, the maximum possible annual profit.
- Justify, also using calculus, that the profit you have found is a maximum.