00:01
In this question, we are given a function g, and we are asked to find its absolute maximum and its absolute minimum values over the given closed interval.
00:09
The only points where absolute maximum and absolute minimum can occur are either the endpoints, in our case, x equals negative 4 and x equals 2, or the critical points.
00:27
And the critical points are points where either g prime of x equals 0 or g prime of x is undefined.
00:47
But first let's calculate g prime of x.
00:51
By the chain rule, g prime of x equals to 2x over x squared minus 1 to the power of 1 3rd.
01:05
Now, g prime of x equals 0 if the numerator is 0, if 2x is 0, and this means that x must be equal to 0.
01:23
So when you plug in x equals 0 in the expression for g prime, you're going to get 0.
01:29
You're going to get 0 over negative 1 to the 1 3rd, but it's going to be 0.
01:38
Now, g prime of x is undefined if the denominator equals to 0.
01:48
And the denominator is x squared minus 1 to the power of 1 third.
02:02
We can erase both sides to the third power.
02:05
We are going to get x squared minus 1 equals to 0, x squared equals to 1, and this gives us two solutions, x equals to negative 1 or x equals to 1.
02:18
So these are all critical points of the function g.
02:27
Also note that all points belong to this interval.
02:31
0 is inside the interval, negative 1 and 1...