00:01
So for this problem, in order to figure out the graph of our equation, we're going to want to start by finding our local extrema.
00:09
So we can go ahead and do that by finding the first derivative of our function.
00:13
So the derivative of x cubed is 3x squared, minus the derivative of 9x is just 9.
00:19
And then 6 is a constant, so the derivative of that will just be 0.
00:23
And so in order to figure out our local extrema, we're just going to solve for where the derivative is equal, or where the derivative is equal.
00:31
To 0.
00:32
So when we solve for x, we find that this is going to occur at x is equal to plus or minus the square root of 3.
00:39
And so to determine where it's going to be increasing and decreasing, i'm going to make a number line.
00:46
So we want to test one point to the left of x is equal to negative 3, one in between these two points, and one to the right of x is equal to the positive square root of 3.
00:57
So one number that we know will be in between the two of them is 0.
01:01
So we we can go ahead and plug that in.
01:03
And we find y prime is equal to 3 times 0 minus 9.
01:08
So this is negative 9, which means that this segment is decreasing.
01:12
And then i am going to wait to test the other ones.
01:16
For now, i'm going to hold off.
01:18
So next we can go ahead and find our inflection points.
01:22
And that's where the second derivative is equal to 0.
01:24
So the second derivative of our function is just 6x.
01:29
So 6x is equal to 0, where x is equal to 0...