High School Calculus/The First Derivative Test

The First Derivative Test edit

The definition of a derivatives tells us that a derivative is the slope of the tangent line at a point on the function.

Derivatives can also tell us if a function is decreasing or increasing at a point.

A function   is increasing on an interval, if for two numbers   and   in the interval   that   is true.

A function   is decreasing on an interval, if for two numbers   and   in the interval   that   is true.


If a function   is continuous on a closed interval   and differentiable on an open interval   then the following applies:

1. If   for all   in   then   is increasing on  

2. If   for all   in   then   is decreasing on  

3. If   for all   in   then   is constant on  


In the last section, we learned about absolute minimums/maximums. Inside a function, other extrema, known as relative extrema, can exist.

The relative extrema of a function are points on a function that are lower or higher than all of the points near them. Such points create "hills" or "valleys" within a given function.

Relative extrema occur at points on a function where the derivative at that point changes from increasing to decreasing, or decreasing to increasing.

If the derivative changes from increasing to decreasing, that point is known as a relative maximum.

If the derivative changes from decreasing to increasing, that point is known as a relative minimum.

By finding the relative extrema of a function, you can then calculate whether or not those extrema are relative minima or maxima using the derivative of the function at those points.

Relative extrema are always critical points of a function.


Example edit

Find the relative extrema of  

First, check if the function is continuous for all  

We can see the function exists for all   therefore, it is continuous.

Second, find the critical numbers of   by using the derivative of the function.

Find the critical numbers by setting  

 

 

 

 

Third, create intervals with your critical numbers.

Since we have two critical numbers, we will have three intervals. They are:

 

Fourth, determine if   is increasing or decreasing over each interval. Do this by evaluating a test number within each interval.

In most cases, it is beneficial to create a table to arrange the present data.

Interval    
Test Value     
Sign of      
Increasing/Decreasing Increasing Decreasing Increasing

Lastly, determine if any relative maximums or minimums are present.

Since   changes from increasing to decreasing to increasing, we can conclude that there is a relative maximum at   and a relative minimum at  

Practice Problems edit

Find the relative extrema of the given functions.