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.
