00:01
In this problem, i'm going to create a little chart, or a little number line here, and i'm going to mark every time the derivative is equal to 0, starting off at x equals negative 2.
00:12
So this is the first derivative, right? then we also have 0, we also have 2, we also have 4, and i don't think we have any other values here.
00:23
And this mark with a positive sign where the derivatives are all the x -axis, and the negative sign is the middle of x -axis.
00:29
So then we start negative, then positive, then negative, then negative, and then positive.
00:36
So now we can ask for question, part a.
00:40
On what intervals is f increasing and what intervals is f decreasing? so f is increasing in the intervals where we have all the positive signs.
00:51
So in this case, you're negative to 0, union, 4, infinity.
00:56
And f is decreasing in the interval, decreasing, where we see all of the negative sides.
01:06
So we're going to start with negative infinity to negative two, union from 0 to 2, union from 2 to 4, and that's pretty much it.
01:15
For part b, you're asked to find for what values of x does f have a local maximum or minimum.
01:21
So where do we have local maximum? so take a look at the little chart that you have here and what this tells us is that here the derivatives went from decreasing to increasing so we expect the function to be something this so we have a local minimum there then here is increasing then decreasing here is decreasing then decreasing and here is decreasing and then increasing so pretty much with that information just let's just put this one more time so this is going to look like this.
01:57
With this information, we expect that f has a local maximum at x equals 0, and f as a local minimum at x equals negative 2, and also at x equals positive 1.
02:27
That's part b.
02:28
Part c now asks us to sketch the graph of the second derivative.
02:35
And here's my attempt to sketch the graph second derivative.
02:39
So again, i can just copy what we have from before, but i'm just going to be careful.
02:46
I'm going to put 0, negative 1.
02:50
So let's see, negative 1 here, negative 2, and here is 1.
02:58
Here's 2, here's 3, and 4, 5, 6, 7, and so on and so on.
03:07
And what do we expect? here we are graphing the derivative of the derivative, the second derivative...