### Video Transcript

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

So weβve been given the graph of
the second derivative of a function and asked to use it to determine something about
the function itself. First, weβll recall that, at an
inflection point, the second derivative π double prime of π₯ is equal to zero. And now, this isnβt a sufficient
condition for a point to be a point of inflection, as itβs 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, letβs 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 weβve 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.