Video Transcript
Find the inflection point on the
graph of 𝑓 of 𝑥 equals 𝑥 cubed minus nine 𝑥 squared plus six 𝑥.
Remember, in order to find the
inflection point on the graph of a function, we need to identify the point at which
the convexity of that function changes. And whilst an inflection point
might appear at a critical point, this is not necessarily the case. We can find the possible locations
of one, though, by using the second derivative. Specifically, if the second
derivative is equal to zero or does not exist, then inflection point might
occur. To be sure, we can check whether
the convexity either side of that point changes. In other words, does the graph
change from being convex up to down or vice versa?
Now 𝑓 of 𝑥 is a polynomial. And this is great because it means
it’s continuous and differentiable over its entire domain. In fact, when we differentiate 𝑓
of 𝑥, we get another polynomial, which is still continuous and differentiable. This means we can begin by finding
the first derivative and then differentiate one more time to get the second
derivative.
We differentiate term by term. The derivative of 𝑥 cubed with
respect to 𝑥 is three 𝑥 squared. Remember, we multiply by the
exponent and then reduce that the exponent by one. Similarly, differentiating negative
nine 𝑥 squared with respect to 𝑥, and we get negative 18𝑥. Finally, the derivative of six 𝑥
is simply six. So the first derivative 𝑓 prime of
𝑥 is three 𝑥 squared minus 18𝑥 plus six.
We’ll repeat this process one more
time to find the expression for the second derivative. This time, we get six 𝑥 minus
18. And to establish where an
inflection point might occur, we’re going to set this equal to zero. In other words, six 𝑥 minus 18 is
equal to zero. To solve for 𝑥, we begin by adding
18 to both sides, so 18 equals six 𝑥 or six 𝑥 equals 18. Then, we divide through by six. So we get 𝑥 equals 18 over six, or
simply three. So we can deduce that an inflection
point might occur at the point where 𝑥 equals three. This is the point where the second
derivative is equal to zero.
To confirm that this is indeed an
inflection point, we’ll look at the second derivative either side of 𝑥 equals three
and check that the convexity does indeed change. Specifically, let’s check the
points where 𝑥 equals two and 𝑥 equals four. Substituting 𝑥 equals two into our
expression for the second derivative, and we get six times two minus 18, which is
negative six. We know that if the second
derivative is negative, the slope of the function is decreasing. This means over the interval where
the second derivative is negative, the function is convex up. So our function is convex up at the
point where 𝑥 equals two.
Repeating this for 𝑥 equals four,
and we get that the second derivative is six times four minus 18, which is positive
six. This time, 𝑓 of 𝑥 is convex down
since the second derivative is positive at 𝑥 equals four. So we observe that the graph of the
function changes from being convex up to convex down about the point 𝑥 equals
three.
We can determine the full
coordinates of this point by substituting 𝑥 equals three into the original
function. 𝑓 of three is three cubed minus
nine times three squared plus six times three, which is equal to negative 36. So the inflection point on the
graph of our function is at three, negative 36.
