00:01
All right, so we got to show that if a is a, that if a is a positive constant, then x equals zero is the only critical value.
00:10
And this is the function, f of x, it's hard to see when i copied it over, f of x equals x plus a square root of x.
00:22
All right, that's the function, i believe.
00:29
Yes, a square root of x.
00:31
There we go.
00:31
All right.
00:33
So a is a positive constant.
00:35
Being a positive constant is important because when we take the derivative, and we're not writing, f prime of x, there we go, is going to be one plus, now if i write it as square root of x as x to the half, i get a over two times x to the negative one half, which puts the square root on the bottom.
01:07
Bottom, square root of x on the bottom.
01:10
So this is my derivative.
01:14
If i set this equal to zero, oops, that's an a on top, not a one.
01:23
If i set this equal to zero, and i'll subtract, i'll subtract the one, so i have negative 1 equals a over two square root of x, and i'll multiply by the two square root of x.
01:39
I have negative two square root of x equals a.
01:43
Well, when i divide by negative 2, i get the square root of x equals negative a over 2.
01:52
Now remember, a is positive.
01:54
A is a positive constant.
01:57
If a was negative, if a was allowed to be negative, then the negative times the negative value would make it positive.
02:13
And we could take the square root of something that would give us a negative answer or give us a positive answer.
02:25
But as it is, there's nothing we can take the square root of to give us a positive answer.
02:31
So can't, come on, ben, can't square root a number to get a negative answer, which is what we're saying, because remember, a is a positive constant.
03:05
A is positive, two is positive on the bottom.
03:08
You divide and that answer is positive.
03:09
But then we got negative in front.
03:12
Okay, but it's saying that zero is the only critical value.
03:17
Remember, you got to find two things.
03:19
We got to find where the derivative is equal to zero, which there isn't anything.
03:27
But then you also got to realize that, well, where is it undefined? f prime of x is undefined at x equals zero because you can't divide by zero.
03:54
Or you can't have a zero in the denominator.
03:58
Okay, right here, right there, we will get a divided by two times zero.
04:05
We'd have a over zero.
04:06
Problem.
04:07
We'd have a problem.
04:08
Okay.
04:09
A is a positive constant, so it can't even be zero.
04:12
It has to be positive.
04:14
Okay.
04:14
So that is why, that's why if a is a positive consonant, the only, the only critical value we get is x equals zero it's because it makes the derivative undefined okay now it says that show that the function is increasing okay so if we want to figure out increasing so we have so here's f prime of x think we're just going to look at the signs we're specifically looking at x is greater than zero i'm assuming but yeah we can't put any numbers and we can't put zero into our derivative it's undefined we can't put negative numbers in because we can't take the square root of a negative so we have to take things that are positive so if i put a number in i'm just going to pick a number like i'll pick this number here four all right so when i take the square root of four i get two and on the bottom i have two times two is four so i'm going to have one plus plus a over 4.
05:25
So this is going to be 1 plus a over 4.
05:28
But remember, a is a positive constant.
05:31
So a positive divided by 4 is still positive...