### Video Transcript

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

We know that π prime of π₯ is actually the slope of π of π₯. If we look at the graph of π of π₯, there are at least two points where the function π of π₯ has a slope of zero. π has a local minimum where π₯ is approximately equal to negative 1.4. And π has a local maximum where π₯ is approximately equal to positive 0.4. We know that, at these two points, the slope π prime of π₯ is equal to zero. So the graph of π 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 π of π₯. So graph a remains a possibility.

Letβs 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 π of π₯, graph b could also be a possibility for our slope of π of π₯. So now, letβs 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 π of π₯. So we can eliminate graph c. Letβs 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 π of π₯. So we can eliminate graph d also.

Weβre left now with graphs a and b as possibilities for the slope of our function π of π₯. We can compare the two graphs to the slope of our function π of π₯ by looking at the direction of the slope either side of the maximum and minimum. If we look at π to the left of negative 1.4, we can see that the direction of the slope of π is negative. The graph of π is sloping downwards. This means that the graph of the slope of π 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, π prime of π₯ is below the π₯-axis for π₯ less than negative 1.4. If we look at graph b, however, π 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 π 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 π of π₯.

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

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