Lesson Video: Interpreting Graphs of Derivatives Mathematics • 12th Grade

Video Transcript

In this video, we will see how we can connect the graph of a function to the graphs of its first and second derivatives. We’ll see how to use the graphs of both the first and second derivatives of a function to make deductions about the graph and properties of the function itself. You should already be familiar with key features of the graph of a function such as local minima and local maxima. You should also be familiar with the definition of concavity of a function and its relationship to the inflection points of a function. Finally, you should be familiar with what it means for a function to be increasing or decreasing on a particular interval, although each of these concepts will be briefly recapped in the context of examples.

Let’s begin by considering a function 𝑓 of 𝑥 equals 𝑥 cubed plus three 𝑥 squared minus nine 𝑥. We can use differentiation to find its first derivative, 𝑓 prime of 𝑥 is equal to three 𝑥 squared plus six 𝑥 minus nine, and also its second derivative, 𝑓 double prime of 𝑥, which is equal to six 𝑥 plus six. Now, let’s sketch the graphs of each of these functions, perhaps using a graphical calculator to help if necessary and then consider what they tell us. Here are those three graphs. The graph of 𝑦 equals 𝑓 of 𝑥 is a cubic. The graph of 𝑦 equals 𝑓 prime of 𝑥 is a quadratic. And the graph 𝑦 equals 𝑓 double prime of 𝑥 is a straight line. From the graph of 𝑦 equals 𝑓 of 𝑥 first of all, we see that our function has two critical points which occur at the 𝑥-values of negative three and positive one.

On the graph of our first derivative, 𝑓 prime of 𝑥, we can see that the value of 𝑓 prime of 𝑥 is zero at each of these 𝑥-values as the line crosses the 𝑥-axis at these two points. We already know from our definition of critical points that 𝑓 prime of 𝑥 is equal to zero or is undefined at the critical point to the function. But from looking at the graph of the first derivative alone, we could have deduced that the function 𝑓 of 𝑥 would have critical points. So that’s local maxima, local minima, or points of inflection at these two 𝑥-values.

Another property of our function 𝑓 of 𝑥 that we can see on its graph is that it is, for example, increasing on the open interval negative infinity, negative three. By considering the graph of 𝑓 prime of 𝑥, we see that 𝑓 prime of 𝑥 is always positive on this interval because the graph of 𝑓 prime of 𝑥 is above the 𝑥-axis. Hence, by considering the sign of 𝑓 prime of 𝑥, that is, whether the line is above or below the 𝑥-axis on a particular interval, we can deduce whether a function is increasing or decreasing on that same interval. We can, therefore, make these deductions about whether a function is increasing or decreasing from the graph of its first derivative without needing to sketch the graph of the function itself.

Further, we see that from the graph of 𝑓 of 𝑥, it appears that 𝑓 has an inflection point at 𝑥 equals negative one, as here the concavity of the graph changes from concave downward to concave upward. By looking at the graphs of both 𝑓 prime of 𝑥 and 𝑓 double prime of 𝑥, we can see that two things also occur at this point. Firstly, the slope of the graph of 𝑓 prime of 𝑓 exchanges from negative to positive. Or we can say that the first derivative changes from decreasing to increasing around 𝑥 equals negative one. This means that the second derivative also changes from negative to positive at 𝑥 equals negative one, which is consistent with what we see in the third graph. The pink line is below the 𝑥-axis for 𝑥-values less than negative one and above the 𝑥-axis for 𝑥-values greater than negative one.

At 𝑥 equals negative one itself, the line intersects the 𝑥-axis giving 𝑓 double prime of negative one is equal to zero. This, combined with the change in sign of 𝑓 double prime of 𝑥, our second derivative, would have been enough for us to deduce that the graph of 𝑓 has an inflection point 𝑥 equals negative one without needing to draw the graphs of either 𝑓 of 𝑥 or 𝑓 prime of 𝑥. Finally, we can also use the graph of our second derivative to classify the turning points of our function 𝑓. We see that the value of the second derivative at 𝑥 equals negative three is negative. And therefore, by the second derivative test, the critical point at 𝑥 equals negative three is a local maximum, which is consistent with what we see on the graph of the function 𝑓.

The value of 𝑓 double prime of one, however, is positive. So by the second derivative test, we know that the critical point at 𝑥 equals one is a local minimum which is again consistent with what we see on the graph of 𝑓. Let’s now consider how to apply some of the general principles we’ve discussed to some examples.

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.

In our next example, we’ll use the graph of a function itself to determine the sign of both its first and second derivatives.

The graph of a function 𝑦 equals 𝑓 of 𝑥 is shown, at which point are d𝑦 by d𝑥 and d two 𝑦 by d𝑥 squared both negative.

So we’ve been given the graph of a function itself. And we’re asked to use this to determine at which of these five points both the first and second derivatives of the function are negative. First, let’s consider the sign of the first derivative d𝑦 by d𝑥 at each point. Recall that the first derivative of a function at a point gives the slope of the tangent to the curve at that point. So by sketching in tangents to the curve at each point, we can determine the sign of their first derivative.

We see that at point 𝐴 the tangent is sloping downwards. So the first derivative, d𝑦 by d𝑥, is indeed negative at point 𝐴. However, at points 𝐵 and 𝐶, the tangents are each sloping upwards, which tells us that the first derivative d𝑦 by d𝑥 will be positive at both 𝐵 and 𝐶. At point 𝐷, the tangent to the curve is horizontal. So the first derivative will be equal to zero, not negative at this point. Finally, at point 𝐸, we see that the tangent is sloping downwards. So the first derivative will also be negative at point 𝐸. We’re, therefore, left with only two options 𝐴 and 𝐸. Next, we need to consider the sign of the second derivative at each of these points. And this is linked to the concavity of the curve at each point.

Recall that the curve is said to be concave downward on a particular interval, if the tangents to the curve in that interval lie above the curve itself. We can also see that when a curve is concave downwards, the slope of its tangent is decreasing. And therefore, its first derivative is also decreasing. When a function is decreasing, then its derivative is negative. And as the derivative of the first derivative is the second derivative, it follows that d two 𝑦 by d𝑥 squared will be less than zero when a curve is concave downward.

This isn’t required here, but a curve is said to be concave upward when the reverse is true. The tangents to the graph lie below the graph itself. The first derivative is increasing and therefore, the second derivative is positive. By considering the graph of 𝑓 of 𝑥, we can see that the tangent we drew at point 𝐸 lies above the curve. And indeed, the shape of the curve is concave downward in this region. However, if we look at point 𝐴, the tangent we drew here is below the curve. And so the graph is concave upward at point 𝐴. This tells us then the second derivative will be negative at point 𝐸, whereas it will be positive at point 𝐴.

We’re left then with only one point at which both the first and second derivatives are negative. It’s point 𝐸. We’ve seen in this example how to determine something about the first and second derivatives of a function from a graph of the function itself.

Now let’s consider how we can determine something about the graph of a function form a graph of its second derivative.

Use the given graph of a function 𝑓 double prime to find the 𝑥-coordinates of the inflection points of 𝑓.

So we’ve been given the graph of the second derivative of a function and asked to use it to determine something about the function itself. First, we’ll recall that, at an inflection point, the second derivative 𝑓 double prime of 𝑥 is equal to zero. And now, this isn’t a sufficient condition for a point to be a point of inflection, as it’s also possible for the second derivative to be zero at a local minimum or a local maximum. But it does give us a starting place. From the given figure, we can see that 𝑓 double prime of 𝑥 is equal to zero in three places, when 𝑥 is equal to one, when 𝑥 is equal to four, and when 𝑥 is equal to seven. So these are the 𝑥 coordinates of the three possible points of inflection of our function 𝑓.

Now, let’s consider a little more about what we know about inflection points. There are points on the graph of a function where its concavity changes either from concave downward to concave upward or vice versa. We also recall that when a function is concave downward, its second derivative, 𝑓 double prime of 𝑥, is negative. And when a function is concave oupward its second derivative is positive. At the inflection point itself, 𝑓 double prime of 𝑥, is equal to zero, which is what we’ve already used to determine our possible points of inflection. But the key point is that when a change in concavity occurs, there will also be a change in the sign of the second derivative. From the given figure, we can see that the sign of the second derivative changes from negative to positive around 𝑥 equals one and changes from positive to negative around 𝑥 equals seven.

However, either side of 𝑥 equals four, the second derivative is positive, and so no change of sign occurs here. Hence, there is no change in the concavity of the function at 𝑥 equals four, but there is at 𝑥 equals one and 𝑥 equals seven. So we can conclude that our function 𝑓 has inflection points at 𝑥 equals one and 𝑥 equals seven.

In our final example, we’ll see how to determine whether a function is increasing or decreasing on a given interval using a graph of its first derivative.

The graph of the derivative 𝑓 prime of a function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

To answer this question, we need to recall the link between whether a function is increasing or decreasing and its first derivative. Formally, a function is increasing on an interval 𝐼 if 𝑓 of 𝑥 one is less than 𝑓 of 𝑥 two for all pairs of 𝑥-values, 𝑥 one and 𝑥 two, with 𝑥 one less than 𝑥 two in the interval 𝐼. In practical terms though, this just means that the graph of the function is sloping upwards. And so its first derivative which, remember, is the slope function of the curve is positive. On the other hand, a function is decreasing on an interval 𝐼 if 𝑓 of 𝑥 one is greater than 𝑓 of 𝑥 two for all 𝑥 one less than 𝑥 two in the interval 𝐼, which in practical terms just means the line is sloping downwards. And so the first derivative, 𝑓 prime of 𝑥, is negative.

To determine the intervals on which any function is increasing or decreasing then, we just need to consider the sign of its first derivative. So the function 𝑓 will be increasing when the graph of its first derivative 𝑓 prime is above the 𝑥-axis. From the given figure, we see that this is true on the open interval, one to five. 𝑓 will be decreasing when the graph of its first derivative is below the 𝑥-axis. From the figure, we see that this is true on two open intervals, the interval zero, one and the interval five, six. So we can conclude then that 𝑓 is increasing on the open interval one to five and decreasing on the open intervals zero to one and five to six.

In this video, we have seen how to use the graphs of the first and second derivatives of a function to make deductions about key properties of the function itself and also how to use the graph of a function to make deductions about its first and second derivatives. We’ve seen that when the first derivative of a function is equal to zero, the function itself has a critical point. And hence, we can determine the 𝑥-values of critical points of the function from a graph of its first derivative. By considering where this graph intersects the 𝑥-axis. We also know that the sign of the first derivative determines whether a function is increasing or decreasing on a given interval. Hence, by considering whether the graph of the first derivative is above or below the 𝑥-axis on a given interval, we can determine whether the function itself is increasing or decreasing.

We’ve also seen that we can use the graph of the second derivative of a function to determine whether the function has inflection points. At points in inflection, the second derivative is equal to zero. But there is also a change in the sign of the second derivative, reflecting a change in the concavity of the function. By considering where the graph of the second derivative is equal to zero and whether it undergoes a change of sign around this value, we can determine the 𝑥-coordinates of any inflection points of the function. Hence, by understanding the links between the graphs of a function and its derivatives, we can deduce key information about the function itself without needing to sketch its own graph. Or we can use the graph of a function itself to determine key properties of its derivatives.