Calculating Maximum Profit | Calculus Optimization Problem

Follow us:

What are optimization problems?

Optimization, by its very name, implies an efficiency of some type: least amount of work, largest volume, smallest distance, greatest profit, etc. If these quantities could be expressed as functions, then graphed, we would be looking for the highest (or lowest) points on the graphs.

Due to the diverse nature of optimization problems, it is difficult to create an algorithm that can be used to solve all problems. They can, generally, be solved by a four step method, outlined as follows:

Step 1: Create a model for the problem.

Define the relationships we are given. Determine what function we are trying to minimize or maximize. This will be the function for which we will have to take the derivative, so we will need to make sure this function has one variable only. Usually this will mean using the given information to make substitutions.

Step 2: Be aware of any constraints on the problem.

We will need to check for boundaries in which our solution may lie. Usually these can be calculated by determining the highest and lowest allowable values for our variable. It is important to note that not all optimization problems have boundaries. In some cases, you may even find it easier to consider these first.

Step 3: Determine the critical points.

Take the derivative of the function and determine the values for which the derivative is 0. Use a sign chart or the second derivative test in order to ascertain whether the critical points are minimum or maximum points.

Step 4: Compare all points.

The function will have a maximum or a minimum at either the endpoints or the critical points. To check, simply compare the values of the function.

- - - - -

A manufacturer of USB storage devices produces up to 100 devices per week. Past experience shows that the manufacturer can sell x devices per week at a price of P(x)=600−3x. The cost of producing x devices per week is C(x)=1000+150x+0.5x^2 dollars.

a) Find the weekly revenue function R(x).
b) Find the weekly profit function P(x).
c) Find the weekly production level to produce a maximum profit.

Recall:

Revenue (R)=number of devices sold∙price per device (P)
Profit (p)=Revenue (R)−cost of device (C)