Closed Interval Method:

Suppose $f(x)$ is a continuous function on an interval $[a,b]$. To find the absolute maximum and minimum valeus of $f(x)$ on $[a,b]$:

1. Find the values of $f$ at all critical point in $(a,b)$
2. Find the values of $f$ at the endpoints $a$ and $b$
3. The absolute maximum of $f$ on $[a,b]$ is given by the largest value from steps (1) and (2), similarly with the absolute minimum

Guidelines:

1. Introduce and name all variables, if possible draw a picture
2. Determine which quantity is to be maximized or minimized in terms of the varibles
3. Write a formula for the quantity in terms of the variables
   1. Formula may include more than one variable
4. Write any equations relating the independent variables (”constraints”) in the formula from step 3
   1. Use these equations to write the quantity to be maximized or minimized as a function of one variable
5. Identify the domain for the function in step 4
6. Apply the Closed Interval Method to find the global maximum or minimum value
   1. Derive the function with respect to the non-constant variable
7. Interpret the real-world significance in terms of the question being asked