- Read the problem.
Draw a picture of the scenario, if you can.
- Label the lengths
or variables in the picture and any variables in the problem.
Reread the problem and write any extra information in the problem
in terms of the variables.
- Decide what you are trying to
optimize (maximize or minimize). Write it as a function of your
variables. We'll refer to this function as f, even though you may
have named it something else.
- Using the relationships between
your variables from the extra information in the problem, rewrite
the function to be optimized in terms of just one variable.
Take the derivative of f.
- Set the derivative of f equal to 0.
We're doing this, because at the relative maxima and minima of f,
the derivative is 0. So if we find where the derivative of f is
0, we'll definitely find these points.
- Solve for the variable
in the equation f'=0. Also, find the values of the variable for
which f' does not exist. All of these points---where the
derivative is 0 and where it doesn't exist---are called critical
- Find f''.
- Check each zero of the derivative to
see if f'' is positive or negative there. If f'' is positive, the
function is concave up at this point, so this zero of f'
corresponds to a relative minimum. If f'' is negative, the
function is concave down at this point, so this zero of f'
corresponds to a relative maximum. If f'' does not exist (as it
will not at each point for which f' doesn't exist) or is equal to
0, look at f' to see if the function was increasing or decreasing
before and after the zero. This information will tell you which
critical points are relative maxima and minima.
- If the
variable was in a bounded range (say it couldn't be less than 0 or
greater than 100), then look at the value of the function at each
critical point of interest---the relative maxima, if you are
trying to maximize; the relative minima, if you are trying to
minimize---and at the end points of the range (0 and 100), and
decide which value optimizes the function.
- Reread the
question to make sure you give the answer requested.