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Find the ordered pair $(x, y)$ on the graph of $y=x^{3}-3 x^{2}$ for which $f^{\prime \prime}(x)=0 .$ (This is called an inflection point.)Show that $\frac{d^{6}\left(x^{6}\right)}{d x^{6}}=6 ! \quad(6 !=6 \times 5 \times 4 \times 3 \times 2 \times 1).$

Calculus 1 / AB

Chapter 23

The Derivative

Section 9

Higher Derivatives

Derivatives

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03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

02:25

For $f(x)=x^{3}-18 x^{2}-1…

09:32

Sketch a graph of the foll…

08:31

01:49

Show that $f(x)=x^{3}-3 x^…

03:49

Identify the coordinates o…

04:16

03:41

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04:26

Okay, So here were given function up, which is gonna be equal to X cube, my ass 18 x squared, minus 10 x plus six. And so we need to find the inflection point and, Graf the function. And so the inflection points occur when the second derivative is equal to zero, we're gonna find the second derivative, and we'll take the first derivative. Hey. And so the derivative of X cubed will be three x squared. The derivative of minus 18. X squared will be minus 36 x and then we'll get a minus 10. And then that, plus sex alternative zero. And now to take thes second derivative taking the derivative of the first primitive, we get six x minus 36. Okay, and this is actually equal to six times X minus six. Okay, so that's gonna be equal to zero. So the second grader, the axe is equal to zero when x is equal to six. So that's where our inflection point will occur. Okay. And if we go ahead and graff this recommend using ah software to get something a little more accurate, but and of course I work. So what? Thio. He says I thought we get a function that looks something like that. Okay. In the Net, X equals six. Here. You would have your inflection point and these air not quite to scale. This is going to be while six is here. Uh, this would be about negative 500.

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