The graph of 𝑓 prime, the derivative of 𝑓, is shown in the figure. At which of the following values of 𝑥 does the function 𝑓 have a point of inflection?
We can see that there are five points labeled on the graph, points 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒. We need to consider what it means for a function to have a point of inflection. A point of inflection is a point on a curve where the concavity of the curve changes either from concave down to concave up, as we see in this sketch, or from concave up to concave down. It’s also true that at a point of inflection the tangent to the curve will cross the curve at that point.
However, this is a graph of 𝑓 prime, the derivative of 𝑓, not a graph of 𝑓 itself, so we need to consider how we can use this graph in order to determine points of inflection on the graph of 𝑓. When a graph is concave up, its derivative will be increasing. And when a graph is concave down, its derivative will be decreasing.
At a point of inflection though, the derivative changes from either increasing to decreasing or decreasing to increasing. And therefore, at a point of inflection, the derivative, that is the first derivative, will have a local minimum or local maximum. This means that at a point of inflection the second derivative, 𝑓 double prime of 𝑥, will be equal to zero.
The second derivative gives the slope of the first derivative. So, what we’re looking for on a graph of the first derivative are points where the slope of the curve is zero. That is, points where the tangent to the curve is horizontal. By visual inspection of the graph, we see that the points 𝑏, 𝑑, and 𝑒 are all local minima or maxima of the graph of 𝑓 prime. And by sketching in the tangents to the graph of 𝑓 prime at these points, we see that they are all horizontal lines. And therefore, the second derivative 𝑓 double prime of 𝑥 will be equal to zero at these three points.
We can see at a glance, but also if we sketch in the tangents to the graph of 𝑓 prime at the points 𝑎 and 𝑐, we see that the tangents are not horizontal at this point. In fact, the point 𝑐 may well be a point of inflection of the graph of 𝑓 prime. But this doesn’t mean it’s a point of inflection of the graph of 𝑓 itself. So, we can conclude that the function 𝑓 has points of inflection at the values 𝑏, 𝑑, and 𝑒 only. As the second derivative of the function is equal to zero at these points.