How to Use Calculus to Solve Optimization Problems (First Derivative Test)

For any continuous function, whenever it is at a maximum or a minimum, the slope of the tangent line will be zero. That means the first derivative of the function at that point is equal to zero. 

Suppose we have 150 feet worth of fencing and we want to use it to build a rectangular enclosure. What is the maximum area we can enclose?

Take a few minutes to try this on your own. Draw a picture. Hint: the four sides of the enclosure must add up to 150. 

Call one side of the enclosure x. The side opposite it will also be x. The other 2 sides will be (150 - 2x)/2 long. Let's add all 4 sides together to double check:

x + x + (150 - 2x)/2 + (150 - 2x)/2 = 2x + 150 - 2x = 150

The area of the enclosure will be y = x*(150 - 2x)/2. We can simplify that to:

y = 75x - x^2

We take the first derivative to get:

y' = 75 - 2x

Then we set y' = 0. 

0 = 75 - 2x

2x = 75

x = 37.5

(150 - 2x)/2 = (150 - 75)/2 = 37.5

The maximum area is enclosed by a square with a side length of 37.5 feet. In general, a square is the geometric figure with 4 sides that encloses the most area with the least perimeter.