Video Transcript
Using the given graph of the function 𝑓, at which values of 𝑥 does 𝑓 have inflection points?
And then we have the graph of the function plotted in the first and fourth quadrant. So to answer this question, let’s remind ourselves what it means for the function to have a point of inflection. An inflection point is a point where the concavity of the graph of the function changes. In other words, the function changes from being concave up to concave down or vice versa. And if we think about the concavity in its most simplest form, we can say the graph is concave down when it looks like a cave and it’s concave up when it looks like a cup.
A little more stringently, though, we say that the graph is concave down over periods where 𝑓 double prime is less than zero, in other words, over intervals where the rate of change of the derivative is negative. Now what this actually means is that the first derivative 𝑓 prime of 𝑥 is actually decreasing over this period. Similarly, the graph is concave up when 𝑓 double prime of 𝑥 is greater than zero, meaning the first derivative, the slope, is increasing over some interval.
So let’s begin by looking at the concavity of the graph. We see that the graph appears to be concave down up to 𝑥 equals three. It’s then concave up between 𝑥 equals three and 𝑥 equals five. And then it’s concave down again for values of 𝑥 greater than five. This means we must have two inflection points. These are at 𝑥 equals three and 𝑥 equals five. So using the given graph of the function 𝑓, we can say that 𝑓 has inflection points at 𝑥 equals three and 𝑥 equals five.
