Video: Finding the Concavity of a Function from Its Derivative Graph

The graph of the first derivative 𝑓′ of a function 𝑓 is shown. On what intervals is 𝑓 concave upward or concave downward?

02:27

Video Transcript

The graph of the first derivative 𝑓 prime of a function 𝑓 is shown. On what intervals is 𝑓 concave upward or concave downward?

Let’s begin by recalling what is meant by these two terms, concave upward and concave downward. If a function is concave upward on a particular interval, then it means that the tangents to the graph of that function all lie below the curve itself on that particular interval. By sketching in these tangents, we can also see that the slope of these tangents is increasing. This is perhaps more obvious on the sketch on the right. But on the sketch on the left we see that the tangents have a negative slope. And they’re becoming less steep, so the values are becoming less negative and therefore increasing.

Hence, we can see a link between the concavity of a function and its first derivative. When a function is concave upward, its first derivative is increasing. If a function is concave downward, however, in a particular interval, it means that the tangents to its graph all lie above the curve itself on that interval. From this sketch, we can see that the slope of the tangent is now decreasing. And hence, we see that when a function is concaved downward, it’s first derivative will be decreasing. This gives us a major clue as to how we can use the given figure, which, remember, is the graph of the first derivative of this function, in order to determine something about the concavity of the function.

To determine where the functions concave upward, we need to see whether graph of the first derivative is increasing, which means it will have a positive slope. We can see that this is true on the open interval zero, one first of all. It’s also true on the open interval two, three and throughout the open interval five, seven. By considering where the slope of our first derivative is negative and hence where the first derivative is decreasing, we can deduce where the function 𝑓 is concave downward. Firstly, the open interval one, two; the open interval three, five; and finally the open interval seven, nine. And so we have our answer to the problem.

We must be careful and clear on what we’re looking for. We’re not looking for where the first derivative is either positive or negative, but rather increasing or decreasing, which is determined not by the sign of the first derivative but by the slope of its graph.

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