00:01
In this problem we have a function and they want firstly to for us to compute the first derivative and the second derivative.
00:08
So the first derivative of the function is using the power rule 3x squared plus 2 times 3 is 6x and the derivative minus 5 is 0.
00:20
So that's the first derivative and the second derivative is the derivative of the first derivative.
00:26
So that's going to be 6x plus 6.
00:31
After that, they want to find what the critical points are.
00:34
The critical points are where the first derivative is zero.
00:38
So all we have to do is write the first derivative equal to zero.
00:44
The easiest way to figure out those values is to factorize this.
00:48
And it has a common factor 3x, factor of x plus 2.
00:59
So for two factors to be zero, either this is zero or this is zero.
01:05
So this can happen to be zero only if x is equal to zero or x equals to minus two.
01:14
So those are the two critical points.
01:18
After that, they want us to find the inflection points.
01:22
Well, the inflection points are where the second derivative is zero as long as is not one of these two critical points.
01:29
So for that, we take this function up here, the second derivative, 6x plus 6, and we equal it to zero.
01:37
And we solve for x.
01:40
6x equals to minus 6.
01:42
X equals to minus 6 over 6.
01:46
So x is equal to minus 1.
01:49
And that would be the inflection point.
01:54
After that in d, they want to find the local maximum and local minimum.
02:00
So those can happen at either of the critical points.
02:07
So what we need to do is figure out, well, x equal to zero, is it a minimum or a maximum? well, we can use the second derivative test.
02:16
The second derivative test, if we value the second derivative at either one of these, it'll tell us if it's a minimum or maximum.
02:25
So if we value it at zero, let's do it at zero.
02:29
The second derivative is this function, so six times zero plus six, which is possible.
02:37
6...