Let 𝑓 be a polynomial function with the values of 𝑓 prime of 𝑥 given for selected values of 𝑥 in the table. Which of the following must be true for 𝑥 between negative three and five?
First of all, note that this is only for selected values of 𝑥. And we’re given five options for our answer. So let’s have a look at what those options mean. 𝑓 is decreasing. When a function is decreasing, its gradient is negative. In other words, 𝑓 prime of 𝑥 is less than zero at each point in the interval, meaning that the function is decreasing on that interval.
𝑓 has at least two relative extrema. At a maximum point, a function goes from increasing to decreasing. And at a minimum point, a function goes from decreasing to increasing. So the slope of the first derivative goes from being positive to negative or negative to positive. So to tell if there is relative extrema, we look for values of 𝑓 prime of 𝑥 changing sign.
𝑓 has no critical points. At critical points, the graph of 𝑓 has a horizontal tangent line. And so 𝑓 prime of 𝑥 equals zero.
The graph of 𝑓 is concave down. When a graph concaves down over an interval, the slope decreases. So all the values of 𝑓 prime of 𝑥 decrease in that interval.
The graph of 𝑓 has at least two points of inflection. A point of inflection is the point at which the curve changes from concave upwards to concave downwards, or vice versa. And so the values of 𝑓 prime of 𝑥 go from increasing to decreasing or decreasing to increasing.
So which of these options must be true? Is 𝑓 decreasing? Well, none of our values for 𝑓 prime of 𝑥 are negative. So we can’t say for definite that 𝑓 is decreasing. Does 𝑓 have at least two relative extrema? Well, do the values of 𝑓 prime of 𝑥 go from positive to negative or negative to positive? Not from the values that we’ve been given. So we cannot say that this must be true.
Is 𝑓 prime of 𝑥 zero for any of our values? None of these values can be critical points. However, there may be other critical points along 𝑓. Remember, we’re only given selected values. So we can’t say for sure that 𝑓 has no critical points. To see if 𝑓 is concave down, we check if 𝑓 prime of 𝑥 is decreasing.
Note that all the values must be decreasing. And we can see by looking at our values for 𝑓 prime of 𝑥 that that is not the case. And finally, does the graph of 𝑓 have at least two points of inflection? Well, we can see a definite increase and decrease in our values of 𝑓 prime of 𝑥. We can see at least two points where this happens. And so it must be true that there are at least two points of inflection. We can say “at least” because we don’t know any other values. There may be more points of inflection.
And so from the selected values that we were given in the question, we can say that it must be true that the graph of 𝑓 has at least two points of inflection.