What is the First Derivative Test for in Calculus?

The first derivative test is one of the most powerful tools that you need to know about when studying calculus. This incredible test has the power to tell you a wide variety of important information about the graph of a function. But what is the first derivative test, and can it really be used to find the maximum points and minimum points of a function?

Let’s find out!

What is the First Derivative Test?

The first derivative test is a process that helps identify the local maximum and local minimum of a given function. These critical values on a function’s graph are also referred to as local extrema. This is just a complex way of saying they are the highest or lowest points on a function’s graph.

The first derivative test also tells us when a given function is increasing and when the function decreases. These are commonly referred to as the intervals of increase and decrease.

The first derivative test works by identifying a specific point on a function’s graph where the function changes from increasing to decreasing, or decreasing to increasing. We call these specific points the critical points of the function.

The First Derivative Test for Maxima and Minima

Depending on the function’s behavior, the critical point of a function can be either a local maximum value or a local minimum value. The term local maxima is a plural version of the word maximum for a specific region on the graph. The term local minima is plural for the minimum of a specific region on the graph of the function.

So how can we use the first derivative test to help us classify critical points as either local maxima or local minima? The key is looking at the sign of the first derivative on either side of the critical points of the function.

Let’s explore this a little bit further!

When Does a Function Have Local Maxima?

A function has local maxima whenever the graph of the function changes from increasing to decreasing. If the function is increasing on the left side of a critical point and decreasing on the right side, we say the critical point is a local maximum.

For example, consider a critical point c. If f'(x)>0 on the left side of c and f'(x)<0 to the right of the critical point c, we say that there is a local maximum at c. Note that the sign of the derivative changes from positive to negative on either side of the critical point. If the slope of the function changes from a positive slope to a negative slope at c, we say that there is a local maximum at c.

When Does a Function Have Local Minima?

By contrast, a function has local minima whenever the graph of the function changes from decreasing to increasing. If the function is decreasing on the left side of a critical point and increasing on the right side, we say the critical point is a local minimum.

For a critical point c, if f'(x)<0 on the left side of c and f'(x)>0 on the right of the critical point c, we say that there is a local minimum at c. Note that the sign of the derivative changes from negative to positive as the function moves past the critical point. If the slope of the function changes from a negative slope to a positive slope at c, we say that there is a local minimum at c.

First Derivative Test Steps

So after asking what is the first derivative test, you now have an understanding of just what this test does. But how does it work? How can we actually apply the first derivative test in practice to help us classify a critical point of a function?

In order to classify a critical point as either a local maximum or a local minimum, we use the first-derivative test by following these steps for a function, f(x).

Step 1: Compute the First Derivative

The first step is to begin by determining the first derivative of the function. This should make sense since we know that we need to examine the sign of the first derivative later on in the process. For a given function, f(x), the first derivative will be f'(x).

Step 2: Set the First Derivative Equal to Zero

The next step is the most important for your understanding of how the first derivative test works. We set the first derivative equal to zero in order to find our test points for the first derivative test.

But why are we interested in when f'(x)=0? 

Remember that the first derivative tells us the slope of the tangent line at any given point on the function. If the first derivative is equal to zero, the slope of the tangent line is equal to zero. This only happens at a maximum or a minimum where the sign of the first derivative changes from either positive to negative or negative to positive.

You can learn more about tangent lines by checking out my instantaneous rate of change guide!

The x-value(s) that you find in this step will be your critical numbers. Critical numbers are numbers that have not yet been classified as maximum or minimum values. During this step you may have to perform some algebra in order to solve for the critical numbers!

Step 3: Find the Intervals of Increase and Decrease

In this next step, we use a test value on either side of the critical number to see what happens to the sign of the first derivative. It is helpful to use one test value to the immediate left and another to the immediate right. This will tell you what the tangent lines look like on either side of the critical number.

It is helpful to summarize this step in a number line, chart, or table. You will see an example of a first derivative test chart in the example below.

Step 4: Determine and Classify Local Extrema

Once you have completed the method of the previous section, you will have a nice summary of the behavior of the graph on either side of the critical points.

The next step is to substitute the critical numbers into the original function in order to arrive at a critical point. Remember that points have both an x and y value!

It is important at this step to classify the local extrema as either local maxima or local minima.

Let’s take a look at the following example to see how to use the first derivative test in action!

First Derivative Test Example

Consider the function \(f(x) = 3x^4+8x^3-18x^2+6\). Use the first derivative test to find all local extrema. 

Step 1: Compute the First Derivative

Since we are working with a polynomial function, we can apply the power differentiation rule to find the first derivative of the given function.

$$ \begin{split} f(x) &= 3x^4+8x^3-18x^2+6 \\ \\f'(x) &= \frac{d}{dx}(3x^4+8x^3-18x^2+6) \\ \\ f'(x) &= 12x^3+24x^2-36x \end{split}$$

Therefore, the first derivative of the function is \(f'(x) = 12x^3+24x^2-36x\).

Step 2: Set the Derivative Equal to Zero

In this next step, we set the derivative equal to zero in order to solve for the critical numbers. In this case, we apply trinomial factoring to help us solve for x.

$$ \begin{split} f'(x) &= 12x^3+24x^2-36x \\ \\0 &= 12x^3+24x^2-36x \\ \\ 0 &= 12x(x^2+2x-3) \\ \\ 0 &= 12x(x+3)(x-1) \end{split}$$

From this point, we set each set of brackets equal to zero to arrive at x = -3, x = 0, and x = 1 as our critical numbers.

Step 3: Find the Intervals of Increase and Decrease

Our next step is to set up a table (or a number line) in order to check the sign of the first derivative on either side of the critical numbers. We do this using a test value on either side of each critical number.

We place a minus signs wherever the slope of the tangent line is negative, and a plus sign wherever the slope of the tangent line is positive. The final row of this table helps us visualize the shape of the graph.

Step 4: Determine and Classify Local Extrema

Using the table above, we can see that the x = -3 is a local minimum, x = 0 is a local maximum, and x = 1 is a local minimum. 

Step 5: Determine Critical Points

The last step is to sub these values into the original function in order to determine the critical points. Doing this results in (-3, -129), (0, 6), and (1, -1) as the critical points!

Applying the First Derivative Test

Now that you have an understanding of just what the first derivative test is and how to use it, you are ready to start applying it to solve problems! 

There are a wide variety of calculus word problems that ask you to find a highest value or lowest value. Optimization problems in particular are specific problems that you will solve that involve finding the extreme points.

When you first encounter the first derivative test, you may start to question just how hard is calculus! The test is can be quite long to apply which feels intimidating for many math students.

Using the previous example, the steps above, and a little bit of practice is all you need to master this powerful calculus tool!

Did you find this guide to the first derivative test for calculus helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful calculus content!

---