Video Transcript
The graph of 𝑓 prime, the
derivative of the function 𝑓, is shown. Which of the following statements
must be true? I) 𝑓 has a relative maximum at 𝑥
equals negative one. II) The graph of 𝑓 has a point of
inflection at 𝑥 equals negative two. III) The graph of 𝑓 is concave
down for zero is less than 𝑥 is less than four.
Let’s take each of these statements
in turn. In statement one, we’re concerned
about a relative maximum. Now, a relative maximum is an
example of a critical point. And we know that the critical
points of a function occur when its first derivative, 𝑓 prime of 𝑥, is equal to
zero or is undefined. From the graph, we see that 𝑓
prime of negative one is indeed equal to zero, because the graph of 𝑓 prime crosses
the 𝑥-axis at this point. And therefore, the function does
have some kind of critical point at 𝑥 equals negative one.
In order to determine what type of
critical point our function has, we need to consider the sign of the slope of the
function either side of the critical point. We see that the graph of the first
derivative is above the 𝑥-axis for 𝑥-values to the left of negative one. And it’s below the 𝑥-axis for
𝑥-values to the right of negative one. The slope therefore changes from
positive to negative, which confirms that this critical point is indeed a relative
maximum. Statement I) is therefore true. Let’s consider statement II), which
concerns points of inflection.
A point of inflection is a point on
a curve where its concavity changes, either from concave upwards to concave
downwards or vice versa. It’s also the case that, at points
of inflection, the second derivative of the function 𝑓 double prime of 𝑥 will be
equal to zero. So in order to determine whether
our function has a point of inflection at 𝑥 equals negative two, we need to
consider 𝑓 double prime of negative two. We need to recall that the second
derivative gives the slope of the first derivative. So by sketching in a tangent to the
graph of 𝑓 prime at 𝑥 equals negative two, we can consider its slope. And hence the value of the second
derivative. We see that the tangent is not
horizontal. And it has a negative slope. And therefore, it isn’t the case
that 𝑓 double prime of negative two is equal to zero. The second statement is therefore
false although it does look like the graph of 𝑓 prime could have a point of
inflection at 𝑥 equals negative two, but not the graph of 𝑓 itself.
Finally, let’s consider statement
III). This statement concerns the
concavity of the curve and in particular whether the graph of 𝑓 is concave down in
a particular interval. We recall that a graph is concave
down in a particular interval if the tangents to the curve lie above the curve
itself in that interval. We also recall that when a graph is
concave down, the slope of its tangent is decreasing. And therefore, the value of the
first derivative 𝑓 prime of 𝑥 will also be decreasing. From the graph of 𝑓 prime, we see
that whilst this might be true for 𝑥-values between zero and one, it isn’t true for
𝑥-values between one and four, because in this interval, the graph of 𝑓 prime is
climbing, which means that the value of 𝑓 prime of 𝑥 is increasing. This tells us then that the graph
of 𝑓 is not concave down on the whole interval zero is less than 𝑥 is less than
four. And so statement III) is false.
We could answer the question then
by concluding that the only one of the three statements which must be true is
statement I).
