Video Transcript
In this video, we will see how we
can connect the graph of a function to the graphs of its first and second
derivatives. We’ll see how to use the graphs of
both the first and second derivatives of a function to make deductions about the
graph and properties of the function itself. You should already be familiar with
key features of the graph of a function such as local minima and local maxima. You should also be familiar with
the definition of concavity of a function and its relationship to the inflection
points of a function. Finally, you should be familiar
with what it means for a function to be increasing or decreasing on a particular
interval, although each of these concepts will be briefly recapped in the context of
examples.
Let’s begin by considering a
function 𝑓 of 𝑥 equals 𝑥 cubed plus three 𝑥 squared minus nine 𝑥. We can use differentiation to find
its first derivative, 𝑓 prime of 𝑥 is equal to three 𝑥 squared plus six 𝑥 minus
nine, and also its second derivative, 𝑓 double prime of 𝑥, which is equal to six
𝑥 plus six. Now, let’s sketch the graphs of
each of these functions, perhaps using a graphical calculator to help if necessary
and then consider what they tell us. Here are those three graphs. The graph of 𝑦 equals 𝑓 of 𝑥 is
a cubic. The graph of 𝑦 equals 𝑓 prime of
𝑥 is a quadratic. And the graph 𝑦 equals 𝑓 double
prime of 𝑥 is a straight line. From the graph of 𝑦 equals 𝑓 of
𝑥 first of all, we see that our function has two critical points which occur at the
𝑥-values of negative three and positive one.
On the graph of our first
derivative, 𝑓 prime of 𝑥, we can see that the value of 𝑓 prime of 𝑥 is zero at
each of these 𝑥-values as the line crosses the 𝑥-axis at these two points. We already know from our definition
of critical points that 𝑓 prime of 𝑥 is equal to zero or is undefined at the
critical point to the function. But from looking at the graph of
the first derivative alone, we could have deduced that the function 𝑓 of 𝑥 would
have critical points. So that’s local maxima, local
minima, or points of inflection at these two 𝑥-values.
Another property of our function 𝑓
of 𝑥 that we can see on its graph is that it is, for example, increasing on the
open interval negative infinity, negative three. By considering the graph of 𝑓
prime of 𝑥, we see that 𝑓 prime of 𝑥 is always positive on this interval because
the graph of 𝑓 prime of 𝑥 is above the 𝑥-axis. Hence, by considering the sign of
𝑓 prime of 𝑥, that is, whether the line is above or below the 𝑥-axis on a
particular interval, we can deduce whether a function is increasing or decreasing on
that same interval. We can, therefore, make these
deductions about whether a function is increasing or decreasing from the graph of
its first derivative without needing to sketch the graph of the function itself.
Further, we see that from the graph
of 𝑓 of 𝑥, it appears that 𝑓 has an inflection point at 𝑥 equals negative one,
as here the concavity of the graph changes from concave downward to concave
upward. By looking at the graphs of both 𝑓
prime of 𝑥 and 𝑓 double prime of 𝑥, we can see that two things also occur at this
point. Firstly, the slope of the graph of
𝑓 prime of 𝑓 exchanges from negative to positive. Or we can say that the first
derivative changes from decreasing to increasing around 𝑥 equals negative one. This means that the second
derivative also changes from negative to positive at 𝑥 equals negative one, which
is consistent with what we see in the third graph. The pink line is below the 𝑥-axis
for 𝑥-values less than negative one and above the 𝑥-axis for 𝑥-values greater
than negative one.
At 𝑥 equals negative one itself,
the line intersects the 𝑥-axis giving 𝑓 double prime of negative one is equal to
zero. This, combined with the change in
sign of 𝑓 double prime of 𝑥, our second derivative, would have been enough for us
to deduce that the graph of 𝑓 has an inflection point 𝑥 equals negative one
without needing to draw the graphs of either 𝑓 of 𝑥 or 𝑓 prime of 𝑥. Finally, we can also use the graph
of our second derivative to classify the turning points of our function 𝑓. We see that the value of the second
derivative at 𝑥 equals negative three is negative. And therefore, by the second
derivative test, the critical point at 𝑥 equals negative three is a local maximum,
which is consistent with what we see on the graph of the function 𝑓.
The value of 𝑓 double prime of
one, however, is positive. So by the second derivative test,
we know that the critical point at 𝑥 equals one is a local minimum which is again
consistent with what we see on the graph of 𝑓. Let’s now consider how to apply
some of the general principles we’ve discussed to some examples.
The graph of the first derivative
𝑓 prime of a function 𝑓 is shown. On what intervals is 𝑓 concave
upward or concave downward?
Let’s begin by recalling what is
meant by these two terms, concave upward and concave downward. If a function is concave upward on
a particular interval, then it means that the tangents to the graph of that function
all lie below the curve itself on that particular interval. By sketching in these tangents, we
can also see that the slope of these tangents is increasing. This is perhaps more obvious on the
sketch on the right. But on the sketch on the left we
see that the tangents have a negative slope. And they’re becoming less steep, so
the values are becoming less negative and therefore increasing.
Hence, we can see a link between
the concavity of a function and its first derivative. When a function is concave upward,
its first derivative is increasing. If a function is concave downward,
however, in a particular interval, it means that the tangents to its graph all lie
above the curve itself on that interval. From this sketch, we can see that
the slope of the tangent is now decreasing. And hence, we see that when a
function is concaved downward, it’s first derivative will be decreasing. This gives us a major clue as to
how we can use the given figure, which, remember, is the graph of the first
derivative of this function, in order to determine something about the concavity of
the function.
To determine where the functions
concave upward, we need to see whether graph of the first derivative is increasing,
which means it will have a positive slope. We can see that this is true on the
open interval zero, one first of all. It’s also true on the open interval
two, three and throughout the open interval five, seven. By considering where the slope of
our first derivative is negative and hence where the first derivative is decreasing,
we can deduce where the function 𝑓 is concave downward. Firstly, the open interval one,
two; the open interval three, five; and finally the open interval seven, nine. And so we have our answer to the
problem.
We must be careful and clear on
what we’re looking for. We’re not looking for where the
first derivative is either positive or negative, but rather increasing or
decreasing, which is determined not by the sign of the first derivative but by the
slope of its graph.
In our next example, we’ll use the
graph of a function itself to determine the sign of both its first and second
derivatives.
The graph of a function 𝑦 equals
𝑓 of 𝑥 is shown, at which point are d𝑦 by d𝑥 and d two 𝑦 by d𝑥 squared both
negative.
So we’ve been given the graph of a
function itself. And we’re asked to use this to
determine at which of these five points both the first and second derivatives of the
function are negative. First, let’s consider the sign of
the first derivative d𝑦 by d𝑥 at each point. Recall that the first derivative of
a function at a point gives the slope of the tangent to the curve at that point. So by sketching in tangents to the
curve at each point, we can determine the sign of their first derivative.
We see that at point 𝐴 the tangent
is sloping downwards. So the first derivative, d𝑦 by
d𝑥, is indeed negative at point 𝐴. However, at points 𝐵 and 𝐶, the
tangents are each sloping upwards, which tells us that the first derivative d𝑦 by
d𝑥 will be positive at both 𝐵 and 𝐶. At point 𝐷, the tangent to the
curve is horizontal. So the first derivative will be
equal to zero, not negative at this point. Finally, at point 𝐸, we see that
the tangent is sloping downwards. So the first derivative will also
be negative at point 𝐸. We’re, therefore, left with only
two options 𝐴 and 𝐸. Next, we need to consider the sign
of the second derivative at each of these points. And this is linked to the concavity
of the curve at each point.
Recall that the curve is said to be
concave downward on a particular interval, if the tangents to the curve in that
interval lie above the curve itself. We can also see that when a curve
is concave downwards, the slope of its tangent is decreasing. And therefore, its first derivative
is also decreasing. When a function is decreasing, then
its derivative is negative. And as the derivative of the first
derivative is the second derivative, it follows that d two 𝑦 by d𝑥 squared will be
less than zero when a curve is concave downward.
This isn’t required here, but a
curve is said to be concave upward when the reverse is true. The tangents to the graph lie below
the graph itself. The first derivative is increasing
and therefore, the second derivative is positive. By considering the graph of 𝑓 of
𝑥, we can see that the tangent we drew at point 𝐸 lies above the curve. And indeed, the shape of the curve
is concave downward in this region. However, if we look at point 𝐴,
the tangent we drew here is below the curve. And so the graph is concave upward
at point 𝐴. This tells us then the second
derivative will be negative at point 𝐸, whereas it will be positive at point
𝐴.
We’re left then with only one point
at which both the first and second derivatives are negative. It’s point 𝐸. We’ve seen in this example how to
determine something about the first and second derivatives of a function from a
graph of the function itself.
Now let’s consider how we can
determine something about the graph of a function form a graph of its second
derivative.
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.
In our final example, we’ll see how
to determine whether a function is increasing or decreasing on a given interval
using a graph of its first derivative.
The graph of the derivative 𝑓
prime of a function 𝑓 is shown. On what intervals is 𝑓 increasing
or decreasing?
To answer this question, we need to
recall the link between whether a function is increasing or decreasing and its first
derivative. Formally, a function is increasing
on an interval 𝐼 if 𝑓 of 𝑥 one is less than 𝑓 of 𝑥 two for all pairs of
𝑥-values, 𝑥 one and 𝑥 two, with 𝑥 one less than 𝑥 two in the interval 𝐼. In practical terms though, this
just means that the graph of the function is sloping upwards. And so its first derivative which,
remember, is the slope function of the curve is positive. On the other hand, a function is
decreasing on an interval 𝐼 if 𝑓 of 𝑥 one is greater than 𝑓 of 𝑥 two for all 𝑥
one less than 𝑥 two in the interval 𝐼, which in practical terms just means the
line is sloping downwards. And so the first derivative, 𝑓
prime of 𝑥, is negative.
To determine the intervals on which
any function is increasing or decreasing then, we just need to consider the sign of
its first derivative. So the function 𝑓 will be
increasing when the graph of its first derivative 𝑓 prime is above the 𝑥-axis. From the given figure, we see that
this is true on the open interval, one to five. 𝑓 will be decreasing when the
graph of its first derivative is below the 𝑥-axis. From the figure, we see that this
is true on two open intervals, the interval zero, one and the interval five,
six. So we can conclude then that 𝑓 is
increasing on the open interval one to five and decreasing on the open intervals
zero to one and five to six.
In this video, we have seen how to
use the graphs of the first and second derivatives of a function to make deductions
about key properties of the function itself and also how to use the graph of a
function to make deductions about its first and second derivatives. We’ve seen that when the first
derivative of a function is equal to zero, the function itself has a critical
point. And hence, we can determine the
𝑥-values of critical points of the function from a graph of its first
derivative. By considering where this graph
intersects the 𝑥-axis. We also know that the sign of the
first derivative determines whether a function is increasing or decreasing on a
given interval. Hence, by considering whether the
graph of the first derivative is above or below the 𝑥-axis on a given interval, we
can determine whether the function itself is increasing or decreasing.
We’ve also seen that we can use the
graph of the second derivative of a function to determine whether the function has
inflection points. At points in inflection, the second
derivative is equal to zero. But there is also a change in the
sign of the second derivative, reflecting a change in the concavity of the
function. By considering where the graph of
the second derivative is equal to zero and whether it undergoes a change of sign
around this value, we can determine the 𝑥-coordinates of any inflection points of
the function. Hence, by understanding the links
between the graphs of a function and its derivatives, we can deduce key information
about the function itself without needing to sketch its own graph. Or we can use the graph of a
function itself to determine key properties of its derivatives.
