#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 decr...