Optimizing a Function of Several Variables
Using Partial Derivatives

Local maximums and local minimums of a function of one variable, f(x), are found at critical points (points where f'(x) = 0 or f'(x) is undefined) and at endpoints, as shown in the figure below. Thus, one way to find local maximums and minimums is by looking at critical points and endpoints.

Example:

Suppose that a manufacturer wants to build a box that will hold one cubic foot of sand. The box will be in the shape of a rectangular prism as shown in the figure below. The manufacturer wants to minimize the cost of materials for the box by minimizing the total surface area of all six sides.

Since the total volume is one cubic foot

and

we can write

The two partial derivatives are

So, to find the critical points we must solve the two equations

or

Thus,

and the box should be a cube.

1. Suppose that a manufacturer wants to build a box with an open top that will hold one cubic foot of sand. She wants to minimize the cost of materials by minimizing the total surface area of the box. What dimensions should the box have? answer

2. Suppose that a manufacturer wants to build a box with an open top that will hold one cubic foot of sand. She wants to minimize the cost of materials. Because sand is so heavy the bottom of the box will cost four times as much per square foot as the sides. What dimensions should the box have? answer

3. Find the largest box (in volume) whose total surface area is six cubic feet. answer

4. Find all the critical points of the function
   

   Draw a graph of this function showing all the critical points and describe each one in prose. answer