Respuesta :
Answer:
hence the required minimum production level is 12units
Step-by-step explanation:
Given the cost function expressed as c(x) = x^3 - 24x^2 + 30,000x
The average cost function will be c(x)/x
Dividing the cost function through by x
Average cost function = c(x)/x = x³/x - 24x²/x + 30,000x/x
Average cost function = x²-24x + 30,000
A(x) = x²-24x + 30,000
If the average cost is minimized, hence dA/dx = 0
dA/dx = 2x - 24
0 = 2x - 24
-2x = -24
Divide both sides by -2
-2x/-2 = -24/-2
x = 12
For the second deriviative
d²A/dx² = 2 which is greater than zero
Hence a production level that will minimize the average cost per item of making x items is 12
The production level that will minimize the average cost per item of making x items is x = 12
How to obtain the minimum value of a function?
To find the minimum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.
Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.
For this case, the function in consideration is: [tex]c(x) = x^3 - 24x^2 + 30000x[/tex]
This is cost function for x items' manufacturing.
The average cost per item would be c(x) / x
This gives us a function a(x) as:
[tex]a(x) = c(x)/x = x^2 -24x + 30000[/tex]
For finding the minimum of a(x), getting its first two derivatives as:
[tex]a'(x) = 2x - 24\\a''(x) = 2 > 0[/tex]
Equating first rate to 0,
[tex]a'(x) = 2x - 24 = 0[/tex] critical point.
Since second rate is positive, the point x = 12 is point of minima, and since only one critical point is there, it is the point on which a(x) is minimum.
Thus, the production level that will minimize the average cost per item of making x items is x = 12
Learn more about minima and maxima of a function here:
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