The graphs of 𝑓 prime, 𝑔 prime, 𝑝 prime, and ℎ prime are shown. Which of the functions 𝑓, 𝑔, 𝑝, and ℎ have a relative maximum on the open interval from 𝑎 to 𝑏?
For this question, we’ve been given four graphs. The first thing we should note is that we have not been given the graphs of 𝑓, 𝑔, 𝑝, and ℎ. But rather, we have been given the graphs of their first derivatives with respect to 𝑥. Next, let’s consider what we mean by a relative maximum. Taking 𝑓 of 𝑥 for an example, we would have a relative maximum on the open interval from 𝑎 to 𝑏 when 𝑓 of 𝑥 is at its highest point on this interval.
More formally, we’d have a relative maximum at 𝑥 equals 𝑐 if 𝑓 of 𝑐 is greater than or equal to 𝑓 of 𝑥 for all 𝑥-values in our interval. And of course, here we’re assuming 𝑐 is indeed within the interval. Finally, we make note of the fact that we do have an open interval. And this means that 𝑎 and 𝑏 are not actually included within this interval.
We have taken this into account by saying that 𝑐 is strictly greater than 𝑎 and strictly less than 𝑏. One helpful way that we can think about a relative maximum is by considering what happens to the gradient of our function either side of 𝑐. Here, we have a relative maximum at 𝑥 equals 𝑐. We can observe that as 𝑥 approaches 𝑐 from the left, we have a positive gradient. And as 𝑥 approaches 𝑐 from the right, we have a negative gradient. At the point where 𝑥 is equal to 𝑐, our gradient is equal to zero.
Of course, we would also have a gradient of zero at a relative minimum. However, at a relative minimum, we would observe a negative gradient as 𝑥 approaches from the left and a positive gradient as 𝑥 approaches from the right. We can therefore say that our first step is to look for places where the first derivative of our function is equal to zero. Here we’ve used 𝑓 of 𝑥 as an example. But we’ll be checking 𝑔, 𝑝, and ℎ as well.
On our graphs, the first derivative of our function with respect to 𝑥 is equal to zero, where our curves intersect the 𝑥-axis. We see that this happens once for 𝑓, once for 𝑔, twice for 𝑝, and once for ℎ. Next, we should take note that, for 𝑝 prime and ℎ prime, these intersections occur at the values of 𝑎 or 𝑏.
Now, since we have an open interval, this means that the values of 𝑎 and 𝑏 are not actually included. This means that there are actually no points on our open interval where 𝑝 prime and ℎ prime are equal to zero. This means that neither of them can have a relative maximum on the open interval. And we can hence eliminate them as choices for our question.
For the remaining graphs of 𝑓 prime and 𝑔 prime, we note that they each only have one point where they’re equal to zero. This means that we don’t have to worry about cases where there are two peaks, with one being higher than the other. Instead, we move forward by considering the implications of the shape of our graphs.
We’ve now found the point, which we’ll call 𝑐, where the first derivative of our functions is equal to zero. We now must check that 𝑓 prime of 𝑥 is greater than zero, which is to say we have a positive gradient as 𝑥 approaches 𝑐 from the left. And 𝑓 prime of 𝑥 is less than zero, which is to say we have a negative gradient as 𝑥 approaches 𝑐 from the right.
In both the remaining two graphs, 𝑐 is equal to zero. For our first graph, when 𝑐 is less than zero, 𝑓 prime is negative. And when 𝑐 is greater than zero, 𝑓 prime is positive. In fact, this is the opposite way round to what we want. And so we can eliminate the graph of 𝑓 prime from our question. Finally, for our graph of 𝑔 prime, we see that when 𝑐 is less than zero, we see that 𝑔 prime is positive. And when 𝑐 is greater than zero, 𝑔 prime is negative.
Remember that, in the question, we’re looking at the graph of 𝑔 prime. What we found is that the graph of 𝑔 will have a positive gradient to the left of 𝑐, which is equal to zero, and a negative gradient to the right of 𝑐, which is equal to zero. The point at which this gradient shifts from being positive to negative is indeed when 𝑥 is equal to zero. This means that the graph of 𝑔 has a relative maximum when 𝑥 is equal to zero. And this value is in our open interval from 𝑎 to 𝑏.
Since we eliminated the other three of our graphs and therefore the other three of our functions, we’re then able to make the following conclusion which answers our question. Of the four functions we were given, 𝑔 is the only function with a relative maximum on the open interval from 𝑎 to 𝑏.