### 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, weβll
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 weβll 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 doesnβt guarantee itβs a point of
inflection. Weβre going to double-check the
concavity of the curve either side of this point. Weβll 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. Thatβs 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.