Derivative test – Types, Explanation, and Examples

Derivative tests are applications of derivatives that help us determine whether a critical number is a local maximum or minimum. Learning about the first and second derivative tests will help us confirm a critical number’s nature without graphing the actual function. The first and second derivative tests allow us to determine a function’s potential critical numbers and local extrema. These two tests tell us whether the function is concaving upward or downward within a given interval. By the end of this article, you should be able to:
• Finding the critical numbers of a function using its first derivative.
• Use the two derivative tests to confirm if a critical number is a local extremum.
• Know the difference between the two derivative tests.
These two derivative tests will also lay out the foundation for other applications and properties we’ll learn in our calculus classes.  Hence, this is going to be a discussion filled with concepts and rules. Take notes when possible and apply what you’ve learned in the problems we’ve prepared just for you.

What is the first derivative test?

The first derivative test allows us to find the intervals where the function is increasing or decreasing. This derivative also helps us find the local maxima and minima of a function. According to the first derivative test, when $c$ is a critical number of $f(x)$, we can observe the following:
• When $f’(x)$ switches from negative to positive at $c$, $f(x)$ has a local (or relative) minimum at $(c, f(c))$.
• Similarly, when $f’(x)$ switches from positive to negative at $c$, $f(x)$ has a local (or relative) maximum at $(c, f(c))$.
• If the sign of $f’(x)$ remains the same on both sides of $c$, $f(x)$ has no extremum at $(c, f(c))$.
The graph of $y = 0.5 x^(3)+2 x^(2)-1$ is as shown above. This function has critical numbers at $x = -\dfrac{8}{3}$ and $x = 0$. Let’s take a look at the signs of $f’(x)$ as we approach each critical number from the left and right.
• At $x = -\dfrac{8}{3}$, we can see that $f’(x)$’s sign changes from positive to negative, so we have a local maximum at $x = -\dfrac{8}{3}$.
• Similarly, $f’(x)$’s sign changes from negative to positive at $x =0$, so we have a local minimum at $x = 0$.
These conditions hold true as long as $f(x)$ is a continuous function. We can also confirm local extremums without the graph and we’ll learn more about that in the next section.

How to do the first derivative test?

We will still use the conditions discussed in our earlier discussion of the first derivative test. These steps will guide you in applying this method correctly:
• Find the critical numbers of $x$ by equating $f’(x)$ to $0$.
• Divide the domain of $f(x)$ into smaller intervals containing the critical numbers as bounds.
• Use a test point for each interval and observe the sign of $f’(x)$.
• Use these results and the first derivative test’s conditions to determine whether the function has a local maximum or minimum.