5.9 Connecting a Function, Its First Derivative, and its Second Derivative

In previous guides, we learned all about making conclusions regarding the behavior of a function based on the behavior of its derivatives such as whether the function is increasing or decreasing at a point, concave up or concave at a point, and more! While we mostly focused on algebraically determining the behavior of functions, we can also determine information graphically! The key features of the graphs of $f$, $f’$, and $f’’$ are all related to one another. 🔑

Let’s dive into how we can do that!

📈 Connecting a Function, Its First Derivative, and Its Second Derivative

Given the graphs of $f$, $f'$, and $f''$ or some combination of the three, we can determine information about another much as we did so algebraically. The knowledge you learned in our previous Unit 5 subtopic guides can be carried over to this subtopic—instead of using the equations for $f$, $f'$, and $f''$, you can look at (one of) their graphs and see where the $x$-axis is crossed or where the graph is positive or negative, increasing or decreasing, etc, to infer information about the other graphs.

📉 Trends and Concavity

Here’s a quick summary of what you’ve learned so far in this unit about trends and concavity:

• When a function is increasing, the first derivative will be positive ($>0$).
• When a function is decreasing, the first derivative will be negative ($<0$).
• When a function is concave up, the second derivative will be positive ($>0$) and the first derivative will increase.
• When a function is concave down, the second derivative will be negative ($<0$) and the first derivative will decrease.

👀 Trends and Concavity Graphically

Let’s apply this information to the following graph of a function, $g(x)$.

Image Courtesy of Zweig Media

Taking a look at this graph, we can describe $g'(x)$ between each interval:

• From $(-\infty,-2)$ and $(0.85,2.8)$, $g(x)$ is decreasing, so $g'(x)$ is negative.
• From $(-2, 0.85)$ and $(2.8,\infty)$, $g(x)$ is increasing, so $g'(x)$ is positive.

What about $g''(x)$? Let’s take a look at the concavity of $g(x)$:

• From $(-\infty,-0.5)$ and $(1.5, \infty)$, $g(x)$ is concave up. Therefore, $g''(x)$ is positive and $g'(x)$ is increasing.
• From $(-0.5,1.5)$, $g(x)$ is concave down. Therefore, $g''(x)$ is negative and $g'(x)$ is decreasing.

🔍 Extrema and Points of Inflection

Based on where the graph of the function changes direction and concavity, we can also interpret maximums, minimums, x-intercepts, and points of inflection of the graphs of the first and second derivatives.

• If $f(x)$ has a relative minimum (the graph changes from decreasing to increasing), then $f'(x)$ will change from negative to positive at that point.
• If $f(x)$ has a relative maximum (the graph changes from increasing to decreasing), then $f'(x)$ will change from positive to negative at that point.
• If $f(x)$ has a point of inflection, changing from concave up to concave down, then $f'(x)$ will have a relative maximum and $f''(x)$ will change from positive to negative at that point.
• If $f(x)$ has a point of inflection, changing from concave down to concave up, then $f'(x)$ will have a relative minimum and $f''(x)$ will change from negative to positive at that point.

If we boil this down to two key concepts, realize that:

1. All relative extrema of $f(x)$ are x-intercepts of $f'(x)$.
2. All points of inflection of $f(x)$ are relative extrema of $f'(x)$.

This may seem like a lot, but once you see it in action, it’ll make more sense! ⬇️

👀 Extrema and POIs Graphically 1

Here’s a relatively easy example! The derivative of the differentiable function $f$, $f'$, is graphed.

Image Created with Desmos.

What can we tell about $f$ at the point $x=1.5$ based on the graph of its derivative $f'$?

By looking at the graph of $f'$, we see that $f'$ crosses the x-axis at the point of interest $x=1.5$. It is negative before $x=1.5$ and positive after $x=1.5$. This means that $f$ is decreasing before the point and increasing after it, indicating that the point $x=1.5$ is a relative minimum on the graph of $f(x)$.

The justification we used above to determine the answer is essentially just applying the First Derivative Test but in graphical form! Here’s a quick look at $f(x)$ and $f'(x)$ so you can really see their relationship:

Image Created with Desmos.

👀 Extrema and POIs Graphically 2

Now, let’s take another look at the example before, but focus on relative extrema and points of inflection.

Image Courtesy of Zweig Media.

You’ll notice the following:

• At $x=-2$ and $x=2.8$, $g(x)$ has relative minima. Therefore, $g'(x)$ has x-intercepts at these points and will change from negative to positive.
• At $x=0.85$, $g(x)$ has a relative maximum. $g'(x)$ has another x-intercept, but the opposite is true: $f'(x)$ will change from positive to negative.
• At $x=-0.5$, $g(x)$ has a point of inflection, changing from concave up to concave down. This means $g'(x)$ will have a relative maximum and $g''(x)$ has an x-intercept changing from positive to negative at $x=-0.5$.
• At $x=1.5$, $g(x)$ has an inflection point, but it changes from concave down to concave up. Therefore, $g'(x)$ has a relative minimum and $g''(x)$ has an x-intercept changing from negative to positive at $x=1.5$.

👀 Extrema and POIs Graphically 3

Before you move on to taking a look at graphs yourself, take a look at the following graph of $h(x)$ and think about:

1. What happens to $h'(x)$ at $x=-1.3$, denoted by the red dotted line?
2. What happens to $h'(x)$ at $x=-0.667$, denoted by the black dotted line?
3. What happens to $h''(x)$ at $x=-0.667$?

Image Created with Desmos.

At $x=-1.3$, $h(x)$ has a relative maximum. This tells us that $h'(x)$ will have an x-intercept at this point and change from positive to negative!

At $x=-0.667$, $h(x)$ changes from being concave down to concave up. This tells us that $h'(x)$ will have a relative minimum at this point and $h''(x)$ has an x-intercept, changing from negative to positive.

Take a look at $h(x)$ in blue and $h'(x)$ in green! You can see exactly these trends.

Image Created with Desmos.

And here’s a graph with $h''(x)$ added as well, denoted in purple.

Image Created with Desmos.

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📝Practice Problems

Now it’s time for you to do some practice on your own! These won’t be as tough, they will more generally test your knowledge of these trends.

❓Practice Problems

Question 1:

The second derivative of the differentiable function $f$, $f''$, is graphed.

Image Created with Desmos.

Given that $f'(1)=0$, what can we tell about $f$ at the point $x=1$ based on the graph of its second derivative $f''$?

Question 2:

The second derivative of the differentiable function $f$, $f''$, is graphed.

Image Created with Desmos.

What can we tell about $f$ at the point $x=-2.4$ based on the graph of its second derivative $f''$?

✅ Answers and Solutions

Question 1:

Answer: $f$ has a relative minimum at the point $x=1$.

Solution:

By looking at the graph of $f''$, we see that $f''$ is positive at the point of interest $x=1$. This means that $f$ is concave up at the point. Combined with the fact that $f'(1)=0$, we can apply the Second Derivative Test to conclude that $f$ has a minimum at $x=1.5$.

Question 2:

Answer: $f$ is concave down at the point $x=-2.4$.

Solution:

By looking at the graph of $f''$, we infer that $f''$ is negative at the point of interest $x=-2.4$. This means that $f$ is concave down at the point.

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⭐ Closing

Woah! You made it to the end of this guide. To practice with some of this material, we recommend getting into Desmos and graphing a function, its first derivative, and its second derivative to see the features of each. Good luck! 🍀