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

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

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Video Transcript

The function 𝑓 of π‘₯ is shown in the graph. Which of the following could be the graph of 𝑓 prime of π‘₯.

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

Let’s look now at option b. This graph also crosses the π‘₯-axis at negative 1.4 and positive 0.4. So, with respect to the maximum and minimum of 𝑓 of π‘₯, graph b could also be a possibility for our slope of 𝑓 of π‘₯. So now, let’s consider graph c. Graph c crosses the π‘₯-axis at negative two, negative 0.5, and positive one. But none of these three points match the maximum or minimum of 𝑓 of π‘₯. So we can eliminate graph c. Let’s look at our final graph, graph d. This crosses the π‘₯-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 π‘₯. 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 π‘₯. We can compare the two graphs to the slope of our function 𝑓 of π‘₯ 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 π‘₯ should be negative for π‘₯ less than negative 1.4. And in fact for graph a, this is the case. In graph a, 𝑓 prime of π‘₯ is below the π‘₯-axis for π‘₯ less than negative 1.4. If we look at graph b, however, 𝑓 prime of π‘₯ in graph b is above the π‘₯-axis for π‘₯ less than negative 1.4. This does not match the direction of slope for our function 𝑓 of π‘₯. 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 π‘₯.

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

Graph a could therefore be the graph of 𝑓 prime of π‘₯.

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