High School Calculus/The First Derivative Test

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The First Derivative Test[edit | edit source]

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 | edit source]

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 | edit source]

Find the relative extrema of the given functions.