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.