Differntiation

Differentiation is used to find the following:
  rates of change of functions
  turning points of functions
  maximum points and maximum values of functions
  minimum points and minimum values of functions
  points of inflexion of functions
  the parts of the domain for which a function is increasing
  the parts of the domain for which a function is decreasing.

An efficiency study conducted for the Undercover Farms Cooperative showed that
the number of cases of cucumbers packed by the average worker x hours after
starting work at 8 a.m. is given by the function
f(x) = -x3 + 4x2 + 12x for 0 < x < 4.

The cooperative wishes to validate this function in terms of the time of the
work day for which the efficiency of the staff is increasing as well as the time
of the work day for which the efficiency of the staff is decreasing.
As a young graduate on staff, you are asked to undertake the validation.

Discuss and demonstrate how you would use Differentiation to find
(a) the efficiency of the staff at 2 hours after the start of work
(b) Is the efficiency increasing or decreasing at that time?
(c) the times of the work day ( i.e. values of x) for which the efficiency of
the staff is increasing;
(d) the times of the work day ( i.e. values of x) for which the efficiency of
the staff is decreasing.