# Video: Using a Graph of a Function to Identify Features of the Graph of the Differential of That Function

The function 𝑓(𝑥) is shown in the graph. Which of the following could be the graph of 𝑓′(𝑥). [A] Graph a [B] Graph b [C] Graph c [D] Graph d

03:20

### Video Transcript

The function 𝑓 of 𝑥 is shown in the graph. Which of the following could be the graph of 𝑓 prime of 𝑥.

We know that 𝑓 prime of 𝑥 is actually the slope of 𝑓 of 𝑥. If we look at the graph of 𝑓 of 𝑥, there are at least two points where the function 𝑓 of 𝑥 has a slope of zero. 𝑓 has a local minimum where 𝑥 is approximately equal to negative 1.4. And 𝑓 has a local maximum where 𝑥 is approximately equal to positive 0.4. We know that, at these two points, the slope 𝑓 prime of 𝑥 is equal to zero. So the graph of 𝑓 prime of 𝑥 should cross the 𝑥-axis at these two points. If we look at the first graph a, it does cross the 𝑥-axis at 𝑥 equal to negative 1.4 and 𝑥 equal to positive 0.4. This fits with our function 𝑓 of 𝑥. So graph a remains a possibility.

Let’s look now at option b. This graph also crosses the 𝑥-axis at negative 1.4 and positive 0.4. So, with respect to the maximum and minimum of 𝑓 of 𝑥, graph b could also be a possibility for our slope of 𝑓 of 𝑥. So now, let’s consider graph c. Graph c crosses the 𝑥-axis at negative two, negative 0.5, and positive one. But none of these three points match the maximum or minimum of 𝑓 of 𝑥. So we can eliminate graph c. Let’s look at our final graph, graph d. This crosses the 𝑥-axis in the same place as graph c, negative two, negative 0.5, and plus one. And again, none of these three points match with our function 𝑓 of 𝑥. So we can eliminate graph d also.

We’re left now with graphs a and b as possibilities for the slope of our function 𝑓 of 𝑥. We can compare the two graphs to the slope of our function 𝑓 of 𝑥 by looking at the direction of the slope either side of the maximum and minimum. If we look at 𝑓 to the left of negative 1.4, we can see that the direction of the slope of 𝑓 is negative. The graph of 𝑓 is sloping downwards. This means that the graph of the slope of 𝑓 of 𝑥 should be negative for 𝑥 less than negative 1.4. And in fact for graph a, this is the case. In graph a, 𝑓 prime of 𝑥 is below the 𝑥-axis for 𝑥 less than negative 1.4. If we look at graph b, however, 𝑓 prime of 𝑥 in graph b is above the 𝑥-axis for 𝑥 less than negative 1.4. This does not match the direction of slope for our function 𝑓 of 𝑥. So we can now eliminate graph b. This leaves us with graph a as the only possibility for the slope of our function 𝑓 of 𝑥.

But let’s just finish our analysis by checking the direction of the slope of 𝑓 of 𝑥 in the other regions of the graph. For 𝑥 between negative 1.4 and positive 0.4, the slope of 𝑓 of 𝑥 is positive. Graph a is above the 𝑥-axis for 𝑥 between negative 1.4 and positive 0.4. So this matches the direction of slope of 𝑓 of 𝑥. And finally, for 𝑥 greater than positive 0.4, the slope of 𝑓 of 𝑥 is negative. Again, this matches with graph a. For 𝑥 greater than 0.4, 𝑓 prime of 𝑥 is below the 𝑥-axis and therefore negative. So that the direction of the slope of 𝑓 of 𝑥 matches our graph a.

Graph a could therefore be the graph of 𝑓 prime of 𝑥.