In this section we are going to look at another type of maximum or minimum problem.  Here we will be looking for the largest or smallest values of a function subject to some kind of constraint.  It’s easiest to see how these work with some examples.  

Critical Points play an important role in optimization problems.   Critical points are defined as those points for which: 

f '(x) = 0, or

f '(x) does not exist

If f(x) has a maximum or a minimum at a point c, then c is a critical point. 

Finding largest or smallest  values of a function on it's domain will involve:

1.	finding the values of the function at the critical points on the domain and 
2.	finding the values of the functions at the endpoints of the interval.

Steps for solving optimization problems:

1.	Read the Problem carefully. What is given? What is the quantity to be optimized?
2.	Visualize (Still Picture). Draw a picture. Label any part that may be important. 
3.	Introduce variables. 
4.	Visualize (Video). Make a sequence of sketches for different variable values (either on paper or in your mind) .
5.	Express the quantity to be optimized in terms of the changing variables.
6.	Reduce the variables to a single variable by using relevant information about the variables (physical, geometrical, etc.)
7.	Obtain a function in one variable for the quantity to be optimized.
8.	Find the appropriate domain (physical  and/or mathematical constraints)
9.	Find critical points and endpoints and test for optima.