# Optimization Using the First Derivative Test - Concept

###### Explanation

Some optimization problems use the first derivative test to find an absolute minimum or maximum. **Using the first derivative test** requires the derivative of the function to be always negative on one side of a point, zero at the point, and always positive on the other side. Other methods of solving optimization problems include using the closed interval method or the second derivative test.

###### Transcript

We have another optimization method we need to study, not every situation is going to land itself to the closed interval method so I want to introduce you to the first derivative test for absolute max and min. Here's the idea behind it, let's say you have a function y equals f of x and the function has a positive derivative to the left of some critical point c and a negative derivative to the right. Well then you can summarize that it has an absolute maximum at c. Now it has to have a positive derivative everywhere to the left and a negative derivative everywhere to the right that will give you an absolute maximum.

This idea is summarizing the first derivative test, suppose f is a differentiable on some interval i and suppose c is the only critical point for f on that interval. If f prime is positive for all x less than c so positive on the left and if it's negative on the right then f will have an absolute maximum and if it's negative on the left and positive on the right it's decreasing and then increasing. It's going to have an absolute minimum at x=c we'll use this method in the next few problems.