# Question Video: Finding the π₯-Coordinates of the Inflection Points of a Function from the Graph of Its Second Derivative Mathematics

Use the given graph of a function πβ³ to find the π₯-coordinates of the inflection points of π.

02:13

### Video Transcript

Use the given graph of a function π double prime to find the π₯-coordinates of the inflection points of π.

So weβ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, weβll recall that, at an inflection point, the second derivative π double prime of π₯ is equal to zero. And now, this isnβt a sufficient condition for a point to be a point of inflection, as itβ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 π double prime of π₯ is equal to zero in three places, when π₯ is equal to one, when π₯ is equal to four, and when π₯ is equal to seven. So these are the π₯ coordinates of the three possible points of inflection of our function π.

Now, letβ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, π double prime of π₯, is negative. And when a function is concave oupward its second derivative is positive. At the inflection point itself, π double prime of π₯, is equal to zero, which is what weβ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 π₯ equals one and changes from positive to negative around π₯ equals seven.

However, either side of π₯ 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 π₯ equals four, but there is at π₯ equals one and π₯ equals seven. So we can conclude that our function π has inflection points at π₯ equals one and π₯ equals seven.