Video: Finding the Inflection Point of the Curve of a Polynomial Function

Video Transcript

Find the inflection point on the graph of 𝑓 of 𝑥 equals 𝑥 cubed minus nine 𝑥 squared plus six 𝑥.

If 𝑝 is an inflection point on a continuous function 𝑓, then the second derivative of 𝑓 of 𝑥 is equal to zero or is undefined at this point. And the concavity of the curve changes at this point. To answer this question then, we’ll begin by finding the second derivative of our function. The first derivative is three 𝑥 squared minus two times nine 𝑥 plus six, which simplifies to three 𝑥 squared minus 18𝑥 plus six. The second derivative is six 𝑥 minus 18. We know that there could be a point of inflection when the second derivative is equal to zero. So we’ll set this equal to zero and solve for 𝑥. We add 18 to both sides of our equation and then we divide through by six. And we see the 𝑥 is equal to three.

But just because the second derivative of 𝑓 of three is equal to zero, that doesn’t guarantee it’s a point of inflection. We’re going to double-check the concavity of the curve either side of this point. We’ll check 𝑓 double prime of two and 𝑓 double prime of four. 𝑓 double prime of two is six times two minus 18, which is negative six. And 𝑓 double prime of four is six times four minus 18, which is six. The second derivative of 𝑓 at two is less than zero and at four is greater than zero. The curve is going from concave downwards to concave upwards. So 𝑥 equals three is indeed a point of inflection. Now, we know this; we can substitute 𝑥 equals three into the equation 𝑓 of 𝑥 to find 𝑓 of three. That’s three cubed minus nine times three squared plus six times three, which is negative 36. The inflection point of our function is at three negative 36.