Which of the following graphs
represents the derivative of 𝑓 of 𝑥 equals 𝑥 cubed plus 𝑥? Option (A), (B), (C), (D), or
In this question, we need to
identify which of five given graphs represents the derivative of the given
polynomial function 𝑓 of 𝑥. The easiest way to answer this
question is to note that since 𝑓 of 𝑥 is a polynomial, we can differentiate it
with respect to 𝑥 term by term by using the power rule for differentiation. We multiply by the exponent of 𝑥
and reduce the exponent by one. We get that 𝑓 prime of 𝑥 equals
three 𝑥 squared plus one.
We see that 𝑓 prime of 𝑥 is a
quadratic with positive leading coefficient. Therefore, its graph is a parabola
that opens upwards. We can see that the constant term
is one, so the 𝑦-intercept of the graph must be at one. We can see that this only matches
the graph in option (C).
This is enough to answer the
question. However, we often want to analyze
the derivative of functions we cannot easily differentiate. In cases like this, it is a good
idea to consider the properties of 𝑓 of 𝑥 and see how these properties will affect
the graph of its derivative. So although we can answer this
question by differentiation, let’s try to do this without differentiating the
function 𝑓 of 𝑥.
First, we can see that 𝑓 of 𝑥 is
the sum of two terms, both of which are increasing functions. We know that the sum of increasing
functions is itself increasing. So 𝑓 of 𝑥 is an increasing
function. Since the function is increasing,
we know that its derivative must be nonnegative. Therefore, the graph of the
derivative must not have any section below the 𝑥-axis. This is enough to say that the
answer must be option (C).
We can analyze the derivative
further by factoring 𝑓 of 𝑥. We see that 𝑓 of 𝑥 equals 𝑥
times 𝑥 squared plus one. We know that 𝑥 squared plus one is
positive for any value of 𝑥. So 𝑓 only has a single
𝑥-intercept at the origin. This allows us to sketch the graph
of 𝑓 of 𝑥. It is a positive leading
coefficient cubic with only one 𝑥-intercept at the origin.
Noting that 𝑓 of 𝑥 is increasing
and sketching its graph gives us the following. We see that its slope is always
nonnegative, so the only possible answer is option (C).