Optimization Using the First Derivative Test - Concept
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.
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.