00:01
All right, in your question, we want to first locate critical values for the given function in the interval.
00:06
So we have f prime of x.
00:08
We need to take the derivative to do that.
00:11
That's going to give you 6x squared plus 6x minus 120.
00:16
And that was just taking the derivative by multiplying the power to the front number and then dropping the power by 1.
00:25
And the last one, 1 multiplied 120, making the x to a 0 power, which makes the x go away.
00:34
And the derivative of our constant is 0.
00:37
Now we want to set this equal to 0, the first derivative, and solve this to see what our critical values might be.
00:47
To do that, i'll factor out a 6.
00:49
I'll have x squared plus x minus 20 equals 0.
00:55
We can factor x squared plus x minus 20 into x plus 5 and x minus 4, because 5 multiplies to a negative 4 to make the negative 20, but they add to make the positive 1 in the middle.
01:12
And that leads you to x plus 5 equaling 0 as one of the solutions here, x minus 4 equals 0.
01:20
And then we have x equals negative 5, and x equals 4.
01:27
So critical values of the given function, technically, this is a critical value, it's in the interval, it's just the endpoint of the interval.
01:35
We also have a critical value here at 4.
01:39
So we've located the critical values.
01:43
Now part b, we want to use the first derivative test to see any local or local maximum or minimum values.
01:55
Now technically, i can't do that because we can't see to the left of negative 5.
02:01
We shouldn't because it's outside the interval of the function.
02:05
So when we use the first derivative test, we're only using it on the 4, okay? because if this is a low point or a high point, it will be an absolute extrema, not a local extrema.
02:21
So moving on to the first derivative test, i'm going to test from negative 5 all the way up to 4, and then from 4 to 8.
02:33
We'll plug in test values.
02:35
I'll plug in a 0 here, a 5 here, and we plug them into the first derivative.
02:41
If you put a 0 into the first derivative, it's easier for us to look at the first derivative in factored form to figure these out...