Video: AP Calculus AB Exam 1 β€’ Section I β€’ Part A β€’ Question 7

The graph of 𝑓′ is shown in the figure. Which of the following statements is true? I) The function 𝑓 is decreasing on the interval (βˆ’βˆž, βˆ’2). II) The function 𝑓 is an absolute maximum at π‘₯ = 0. III) The function 𝑓 is a point of inflection at π‘₯ = 2.

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Video Transcript

The graph of 𝑓 prime is shown in the figure. Which of the following statements is true? I) The function 𝑓 is decreasing on the interval negative infinity to negative two. II) The function 𝑓 is an absolute maximum at π‘₯ equals zero. III) The function 𝑓 is a point of inflection at π‘₯ equals two.

Let’s take each of these statements in turn. If a function is decreasing at a particular point or on a particular interval, then its derivative 𝑓 prime must be negative at that point or on that interval. From the figure, we see that the graph of 𝑓 prime is below the π‘₯-axis for all π‘₯-values in the open interval negative infinity to negative two, which means that 𝑓 prime β€” the first derivative of 𝑓 β€” is indeed negative on this interval.

So, the first statement the function 𝑓 is decreasing on the interval negative infinity to negative two is true. The question doesn’t say though that only one of the statements is true. So we need to check the other two. The second statement is that the function 𝑓 has an absolute maximum at π‘₯ equals zero. Now, if we consider the graph of 𝑓 prime, we see that 𝑓 prime is positive when π‘₯ is equal to zero. 𝑓 prime is equal to two.

As the value of 𝑓 prime β€” the first derivative β€” is positive when π‘₯ is equal to zero, this means that the function 𝑓 is increasing at this point. And so, the function can’t have an absolute maximum at π‘₯ equals zero because the function is increasing; it’s getting larger. So the second statement is false.

The third statement is that the function 𝑓 has a point of inflection when π‘₯ is equal to two. Points of inflection are points on a curve at which there is a change in the direction of the curvature, either from concave to convex or vice versa. Points of inflection can also be stationary or critical points of a curve if the first derivative 𝑓 prime is equal to zero at the point of inflection. But they don’t have to be. As long as there’s a change in the direction of the curvature, then it’s a point of inflection, regardless of whether or not it’s also a critical point.

At points of inflection, the second derivative of a function 𝑓 double prime is equal to zero. So how can we use the graph of the first derivative in order to determine whether or not the second derivative is equal to zero when π‘₯ is equal to two? The second derivative is the derivative of the first derivative. That’s the gradient of the first derivative.

So if you want to look for points of inflection, we need to look for points on the graph of the first derivative where the gradient is equal to zero. And specifically, if you want to determine whether the function 𝑓 is a point of inflection at π‘₯ equals two, we need to consider the gradient of the graph 𝑓 prime at this point.

We can do this by drawing a tangent to the graph of 𝑓 prime at the point where π‘₯ is equal to two. And we see that it is a horizontal line. And so, 𝑓 double prime of two is indeed equal to zero. This tells us that the function 𝑓 does have a point of inflection at π‘₯ equals two. So the third statement is also true.

Our answer to the question then which of the following statements is true is one and three only.

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