00:01
We are given the cost function c of x equals 0 .25 x squared plus 700x plus 10 ,000, where x represents the number of items.
00:11
And we have to determine the average cost function.
00:19
The average cost function is given by c bar of x.
00:23
And this equals the cost function over x.
00:28
So let's determine this.
00:29
So let's replace the cost function, which is 0 .25x squared plus 700.
00:36
X plus 10 ,000 and this is over x and here we can divide all the terms in the numerator by x so we do like this point to 5 x squared divided by x this is x squared and then 700x divided by x so the first term is 0 .25x.
01:14
X divided by x is 1.
01:16
So 700 times 1 is 700.
01:19
And then we write this as 10 ,000 over x.
01:25
And so this is the average cost function represented as c bar of x.
01:33
Now we have to determine the number of items where this average cost function is minimum.
01:40
And for that, we are going to set up the equation that is c bar prime of x.
01:51
The derivative of this cost, every cost function is given by c bar prime of x.
01:57
We have to set up this equals zero and solve for x.
02:02
And we will also have to determine the sign chart for the derivative of the average cost function.
02:08
And using the sign chart, we can determine at what value of x, the average cost function is minimal.
02:14
So first, let's determine the derivative of a cost function.
02:18
So i'm going to find the derivative of this c bar of x.
02:22
And this is c bar prime of x.
02:25
This equals 0 .25.
02:28
The derivative of x is 1.
02:30
So this will be 0 .25 times 1 plus the derivative of constant 700.
02:35
This one is 0.
02:37
And then here we can consider this as 10 ,000 times the derivative of 1 over x.
02:44
Or otherwise we can rewrite this in exponent form.
02:48
That is this is x power negative 1.
02:52
We should find its derivative.
02:54
We use power rule to find the derivative of x power negative 1, which equals negative 1, x raised to the power negative 2.
03:02
Or otherwise, we can rewrite this as negative 1 over x squared.
03:07
So therefore, the derivative of 1 over x, we can replace this here as negative 1.
03:15
Over x squared and when you multiply these two we get negative 10 ,000 over x squared.
03:23
Let's write down here.
03:24
This is c bar prime of x equals 0 .25 minus 10 ,000 over x squared.
03:35
So we found the derivative of the average cost function.
03:40
Now we have to set this up equal 0 and solve for x.
03:43
So i'm going to replace c bar prime of x equal 0.
03:46
And this means we will get 0 equals 0 .25 minus 10 ,000 over x squared.
03:58
We'll solve for x from this one by adding 10 ,000 over x squared on both sides.
04:04
So we will get 10 ,000 over x squared equals 0 .25.
04:10
Multiply both sides by x squared.
04:12
We get 0 .25 x squared equals 10 ,000.
04:18
Divide both sides by 0 .25, we get x squared equals 10 ,000.
04:25
This is over x.
04:27
I'm sorry, this is over 0 .25.
04:33
And this equals, let's use a calculator, 10 ,000 divided by 0 .25...