00:01
According to this question, we are given that fx is equals to x2 plus of 3x minus of 18 divided by x plus 3.
00:10
So we need to check that if rolle's theorem can be applied for this given function in the interval minus of 6, 3.
00:18
And we also need to find out find the values of c which is satisfying the conclusion of that theorem.
00:25
So firstly we need to check that if rolle's theorem can be satisfied.
00:28
So for rolle's theorem there are three important things which we need to keep in mind.
00:34
And they are namely the first one is that fx is continuous in the open, it's the closed interval a, b.
00:52
And the second most important point is that fx is differentiable in the open interval that's a, b.
01:03
So i'm not writing the entire statement, i'm just telling you the main points.
01:07
And moreover, if fa is equals to fb, so in that case, there exists a c for which f dash c will be equals to 0.
01:19
So what you need to do is that you will first find out f dash x and you will just equate it to 0.
01:27
So whatever value you are getting, that value will give you that is the constant c because the c is actually lying in between a and b.
01:38
So firstly, we will be checking for f of a and a is minus 6.
01:43
So we need to put x as minus 6 over here in fx.
01:47
So before i just find out this f of minus 6, what i'm going to trying to do is that i'm just trying to simplify this fx.
01:56
So this is a quadratic equation in the numerator.
01:59
So i can just write it like this is x squared plus of 6x minus of 3x and this is minus of 18.
02:05
And this is whole divided by x plus of 3.
02:09
And this is fx is equal to x you can take common you're left with x plus 6 then again you can take minus 3 common you're left with x plus 6.
02:18
And this is whole divided by x plus 3.
02:21
So thus fx is equals to x minus of 3 in multiplication you have x plus 6 and this is whole divided by x plus 3.
02:30
It's not so important just to avoid a bit of tough calculation.
02:36
I've just done this because it will be easier for us to put the values in this case.
02:40
So f of minus 6 means you have to replace x with minus 6 in equation 1.
02:45
So when you do that, this second bracket in the numerator will become 0 and anything multiplied to 0 will give you 0 in the denominator you have in place of x you write minus x 6 plus 3 which will give you 0 divided by minus 3 which is equals to 0.
03:02
So f of minus 6 has come out to be equals to 0.
03:05
Now check for f of 3.
03:07
So i'm replacing x with 3 in equation 1.
03:10
The first bracket in the numerator will be equals to 0 and anything multiplied to 0 gives you the result as 0.
03:16
In the denominator you have 3 plus 3 so you have 0 by 6 which is equals to 0.
03:22
So thus you're able to see that f of minus 6 is equal to f of 3.
03:27
Okay so the conditions are satisfied.
03:30
Moreover fx is also continuous.
03:32
So you write moreover fx is continuous in the closed interval minus of 6, 3 and fx is differentiable in the open interval minus of 6, 3.
03:59
So thus the conditions of rolle's theorem are satisfied.
04:03
So you write thus the conditions of rolle's theorem are satisfied.
04:22
So thus we can f dash x can be equated to 0 to give c.
04:29
So you write f dash x can be equated to 0 to give c.
04:41
That's the most important thing.
04:42
Okay.
04:42
So thus find out f dash x for this.
04:45
We are applying the division rule not to differentiate this fx.
04:50
So i'm just choosing this equation the asterisk marked equation.
04:55
So as per the formula we write the second function.
04:58
Your second function is the one which is written in the denominator.
05:01
So this is your first function.
05:03
All right.
05:03
And this is your second function...