00:01
In this question, we are asked to find the local extremo of the function f.
00:05
But first, we need to find the candidates for a local extremo, which are called critical points.
00:17
And the critical points are either the points where f prime of x equals to zero, or the points where f prime of x is undefined.
00:32
Let's calculate f prime of x.
00:35
F prime of x equals to negative, well, the derivative of negative 1 is 0 because it's a constant.
00:41
Then it's negative 21 halves times 2x minus 7 thirds multiplied by 3x squared and this equals to negative 21x minus 7x squared we can factor out negative 7x to get negative 7x multiplied by x plus 3 all right that's f prime of x note that f prime of x is defined and continues everywhere, which means that the only points, the only place where we can find the critical points are f prime of x equals to zero.
01:46
Let's solve f prime of x equals zero.
01:51
In our case, it's going to be negative 7x times x plus 3 equals 0.
02:03
Therefore, and this function equals 0 if x equals 0 or x equals to negative 3.
02:10
So these are the critical points.
02:13
Or these are the points where the local maxing or minimum values can occur, potentially occur.
02:21
But it's not guaranteed yet, so we have to use the first derivative test now.
02:28
These two critical points, they split the whole number line into three sub -intervals.
02:33
From 0 to infinity, from negative 3 to 0, and from negative infinity to negative to negative 3.
02:52
Now we need to study, determine the sign of f prime on each of this...
