For the function f(x) = 4x^4 - 4x^3 - 4x^2 + 4, find f”(x)....

For the function $f(x) = 4x^4 - 4x^3 - 4x^2 + 4$, find $f''(x)$. Then find $f''(0)$ and $f''(4)$. $f''(x) = 48x^2 - 24x - 8$ Select the correct choice below and fill in any answer boxes in your choice. A. $f''(0) = $ (Simplify your answer.) B. $f''(0)$ is undefined. Select the correct choice below and fill in any answer boxes in your choice. A. $f''(4) = $ (Simplify your answer.) B. $f''(4)$ is undefined.

Transcript

00:01 I'm going to be under the understanding that you have some sort of experience with the chain rule.

00:05 Otherwise, this problem is going to be difficult.

00:09 So anyway, we're looking at 3x plus 1 to the fourth power, and i'll just write off to the side that i'm doing the chain rule.

00:16 And if you made more of an internet search, that might be to your benefit.

00:20 But what the chain rule is, you move that 4 in front, and you subtract 1 from the x1.

00:26 You leave this interfunction alone, and then you multiply by the derivative of the interfirm.

00:31 Function, which is three.

00:33 But then you have to do the chain rule again for the second derivative.

00:38 Now, kind of a heads up, this four times three is 12, but then you have to do move that exponent in front again.

00:47 So times three, you leave that interfunction alone again, and you subtract one from that exponent, but again, you have to multiply by the derivative of the interfunction, which is another three.

00:59 So at this point, you're looking at 12 times 3 is 36 times 3 is 108.

01:06 So let's go ahead and jot that down.

01:13 Under 8, 3x plus 1 squared.

01:17 So i should probably double check.

01:19 I have the right function.

01:20 3x plus 1 to the 4.

01:21 Okay.

01:22 So now you're asked to find the second derivative at 0, which is actually pretty easy.

01:27 So i'll just jot it down right here...

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