00:01
In this question, we are asked to find all a relative extrema of the given function f and classify them as relative maxima and relative minimum.
00:11
Right, to find the relative extrema, the first night need to find the critical points of the function f.
00:17
And to find the critical points, we need to solve the equation f prime of x equals 0.
00:23
In our case, f prime of x equals to 4x cube plus 36 x squared.
00:31
And we want this to be equal to 0.
00:34
We can factor out 4x squared and in parentheses we're going to have x plus 9 equals 0 and this means that the solutions are either x equals 0 or x equals to negative 9.
00:59
Now we are going to use to classify it to figure out if this so these are two possible extreme points of the function f.
01:11
But we don't know yet for sure if they are like extrema.
01:15
To do that to determine if they are indeed like relative extrema, we need to use a second derivative test.
01:23
For the second derivative test, we need to calculate f double prime of x and f double prime of x equals to 12x squared plus 72x.
01:39
And we can factor out 12x.
01:45
In parentheses, we're going to get x plus 6.
01:51
Now, we need to calculate the values of the second derivative at the critical points.
02:00
First, let's calculate f double prime at 0, and f double prime of 0 equals to 0.
02:05
And in this case, this means that the second derivative test is inconclusive.
02:14
We will get back to this case after we are finished with x equals...