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

Find the inflection point on the graph of π(π₯) = π₯Β³ β 9π₯Β² + 6π₯.

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### Video Transcript

Find the inflection point on the graph of π of π₯ equals π₯ cubed minus nine π₯ squared plus six π₯.

If π is an inflection point on a continuous function π, then the second derivative of π of π₯ 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, weβll begin by finding the second derivative of our function. The first derivative is three π₯ squared minus two times nine π₯ plus six, which simplifies to three π₯ squared minus 18π₯ plus six. The second derivative is six π₯ minus 18. We know that there could be a point of inflection when the second derivative is equal to zero. So weβll set this equal to zero and solve for π₯. We add 18 to both sides of our equation and then we divide through by six. And we see the π₯ is equal to three.

But just because the second derivative of π of three is equal to zero, that doesnβt guarantee itβs a point of inflection. Weβre going to double-check the concavity of the curve either side of this point. Weβll check π double prime of two and π double prime of four. π double prime of two is six times two minus 18, which is negative six. And π double prime of four is six times four minus 18, which is six. The second derivative of π at two is less than zero and at four is greater than zero. The curve is going from concave downwards to concave upwards. So π₯ equals three is indeed a point of inflection. Now, we know this; we can substitute π₯ equals three into the equation π of π₯ to find π of three. Thatβ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.