The graph of the function 𝑦 equals 𝑓 prime of 𝑥 is shown. Determine over which intervals the function 𝑓 of 𝑥 is positive.
So we are given the graph of the first derivative of a function, and we’re asked to use it to determine something about the function itself. First, we recall that at a critical point of our function 𝑓 of 𝑥, the first derivative 𝑓 prime of 𝑥 will equal zero. A critical point could be a relative minimum, a relative maximum, or a point of inflection. From the given figure, we can see that 𝑓 prime of 𝑥 equals zero at four different places: when 𝑥 is equal to one, when 𝑥 is equal to two, when 𝑥 is equal to five, and when 𝑥 is equal to seven.
Next, we will recall the first derivative test, which says if 𝑓 prime of 𝑥 is positive on an open interval, then 𝑓 is increasing on that same interval. We’ve highlighted the intervals on the graph of 𝑓 prime of 𝑥 where 𝑓 prime of 𝑥 is positive. This points to 𝑓 increasing on the intervals of negative ∞ to one, two to five, and seven to ∞.
The second part of the first derivative test says that if 𝑓 prime of 𝑥 is negative on an open interval, then 𝑓 is decreasing on that interval. We can highlight those intervals on the graph of 𝑓 prime in blue. This implies that 𝑓 is decreasing on the intervals from one to two and five to seven.
Let’s clear some space and summarize what we have learned so far about the function 𝑓 of 𝑥. From the graph of 𝑓 prime of 𝑥, we have determined that 𝑓 will have critical points at one, two, five, and seven and that it will be increasing on intervals from negative ∞ to one, two to five, and seven to ∞ and decreasing on intervals from one to two and five to seven.
None of this information directly tells us where the function is positive. So let’s sketch an approximate graph of 𝑓 of 𝑥 to see if we can gain any more insights. From negative ∞ to one, our graph is increasing. Then, from one to two, the graph of 𝑓 is decreasing, then from two to five, increasing again, then another decrease from five to seven, and a final increase from seven to ∞. Then, we can place our critical points between the intervals of increasing and decreasing. It appears that we have local maxima at one and five and local minima at two and seven.
In this example, we were asked to find the intervals where the function is positive. But we have no way of determining where the 𝑥-axis is. We may have the intervals for which the function slopes upward or slopes downward and even the 𝑥-values of the local minima and maxima. We may know for what intervals 𝑓 prime is positive, but this does not indicate where the function 𝑓 is positive. We can even sketch a few examples for which 𝑓 is positive on two intervals or maybe one interval, but we really cannot know for sure.
To answer this question with any sort of certainty, we need more information about 𝑓 of 𝑥.