Video: Finding the 𝑥-Coordinates of the Inflection Points of a Function from the Graph of Its Second Derivative

Video Transcript

Use the given graph of a function 𝑓 double prime to find the 𝑥-coordinates of the inflection points of 𝑓.

So we’ve been given the graph of the second derivative of a function and asked to use it to determine something about the function itself. First, we’ll recall that, at an inflection point, the second derivative 𝑓 double prime of 𝑥 is equal to zero. And now, this isn’t a sufficient condition for a point to be a point of inflection, as it’s also possible for the second derivative to be zero at a local minimum or a local maximum. But it does give us a starting place. From the given figure, we can see that 𝑓 double prime of 𝑥 is equal to zero in three places, when 𝑥 is equal to one, when 𝑥 is equal to four, and when 𝑥 is equal to seven. So these are the 𝑥 coordinates of the three possible points of inflection of our function 𝑓.

Now, let’s consider a little more about what we know about inflection points. There are points on the graph of a function where its concavity changes either from concave downward to concave upward or vice versa. We also recall that when a function is concave downward, its second derivative, 𝑓 double prime of 𝑥, is negative. And when a function is concave oupward its second derivative is positive. At the inflection point itself, 𝑓 double prime of 𝑥, is equal to zero, which is what we’ve already used to determine our possible points of inflection. But the key point is that when a change in concavity occurs, there will also be a change in the sign of the second derivative. From the given figure, we can see that the sign of the second derivative changes from negative to positive around 𝑥 equals one and changes from positive to negative around 𝑥 equals seven.

However, either side of 𝑥 equals four, the second derivative is positive, and so no change of sign occurs here. Hence, there is no change in the concavity of the function at 𝑥 equals four, but there is at 𝑥 equals one and 𝑥 equals seven. So we can conclude that our function 𝑓 has inflection points at 𝑥 equals one and 𝑥 equals seven.