Video Transcript
The graph of 𝑓 prime is shown in
the figure. Which of the following statements
is true? I) The function 𝑓 is decreasing on
the interval negative infinity to negative two. II) The function 𝑓 is an absolute
maximum at 𝑥 equals zero. III) The function 𝑓 is a point of
inflection at 𝑥 equals two.
Let’s take each of these statements
in turn. If a function is decreasing at a
particular point or on a particular interval, then its derivative 𝑓 prime must be
negative at that point or on that interval. From the figure, we see that the
graph of 𝑓 prime is below the 𝑥-axis for all 𝑥-values in the open interval
negative infinity to negative two, which means that 𝑓 prime — the first derivative
of 𝑓 — is indeed negative on this interval.
So, the first statement the
function 𝑓 is decreasing on the interval negative infinity to negative two is
true. The question doesn’t say though
that only one of the statements is true. So we need to check the other
two. The second statement is that the
function 𝑓 has an absolute maximum at 𝑥 equals zero. Now, if we consider the graph of 𝑓
prime, we see that 𝑓 prime is positive when 𝑥 is equal to zero. 𝑓 prime is equal to two.
As the value of 𝑓 prime — the
first derivative — is positive when 𝑥 is equal to zero, this means that the
function 𝑓 is increasing at this point. And so, the function can’t have an
absolute maximum at 𝑥 equals zero because the function is increasing; it’s getting
larger. So the second statement is
false.
The third statement is that the
function 𝑓 has a point of inflection when 𝑥 is equal to two. Points of inflection are points on
a curve at which there is a change in the direction of the curvature, either from
concave to convex or vice versa. Points of inflection can also be
stationary or critical points of a curve if the first derivative 𝑓 prime is equal
to zero at the point of inflection. But they don’t have to be. As long as there’s a change in the
direction of the curvature, then it’s a point of inflection, regardless of whether
or not it’s also a critical point.
At points of inflection, the second
derivative of a function 𝑓 double prime is equal to zero. So how can we use the graph of the
first derivative in order to determine whether or not the second derivative is equal
to zero when 𝑥 is equal to two? The second derivative is the
derivative of the first derivative. That’s the gradient of the first
derivative.
So if you want to look for points
of inflection, we need to look for points on the graph of the first derivative where
the gradient is equal to zero. And specifically, if you want to
determine whether the function 𝑓 is a point of inflection at 𝑥 equals two, we need
to consider the gradient of the graph 𝑓 prime at this point.
We can do this by drawing a tangent
to the graph of 𝑓 prime at the point where 𝑥 is equal to two. And we see that it is a horizontal
line. And so, 𝑓 double prime of two is
indeed equal to zero. This tells us that the function 𝑓
does have a point of inflection at 𝑥 equals two. So the third statement is also
true.
Our answer to the question then
which of the following statements is true is one and three only.
