00:01
Okay, so we have the function x to the 4th minus 3x cubed minus 6x squared plus 7x plus 30.
00:09
And the first thing we have to do is graph this on the given window from minus 3 to 5 in the x direction and minus 10 to 50 in the right y direction, which we have shown here.
00:20
The red line is our function f of x.
00:22
So we want to see what the derivative is going to look like for this function.
00:27
If we can approximate this by looking at a couple things.
00:29
The first thing we're going to look at is if there are any horizontal tangent lines.
00:35
So what we mean here, if we see any peaks or cusps or minimum or maximum points in our graph, we can say those are going to have horizontal tangent lines, which means the slope at those specific points are going to be zero.
00:50
So right away, we can see that there will be a horizontal tangent line around here at this point, one here at this other peak, and also another one down here.
01:02
So it's important to note these because these are going to tell us when our graph of the derivative crosses the x -axis, a .k .a.
01:13
When the derivative is equal to zero.
01:15
So this means the derivative is going to be equal to zero at around, let's say, negative 1 .3, if we're going to approximate that, at 0 .5, and at about 3.
01:29
So these are important points to know because this is where we're going to base the rest of our derivative graph.
01:35
And we can see here that for this left portion, the slope of this is decreasing, which gives us a negative value through the slope.
01:47
And this section is positive, because again, it's positive slope, negative slope there.
01:54
And here we have a big chunk of negative slope.
01:57
And then the slope increases again in this final piece here.
02:02
So this can say piece one is a negative slope.
02:08
This piece here positive negative than positive so if you want to approximate this graph we can do so but kind of slightly sketching here so if we have a negative slope and it's becoming so it comes out a steep negative slope then it's coming back to a slope of zero we can kind of draw it as this again this will be pretty small but it's going to be a steep negative then it's going to come back to zero and since we're at zero at this point, and we're transitioning to a positive slope, that's when our slope becomes positive.
02:53
It's becoming less and less positive.
02:55
So it's going to be some type of peak and then come back to this zero point...