Respuesta :
Answer:
A production level that will minimize the average cost of making x items is x=5.
Step-by-step explanation:
Given that
[tex]c(x)=7x^3-70x^2+13,000x[/tex]
is the cost of manufacturing x items
To find a production level that will minimize the average cost of making x items:
The average cost per item is [tex]f(x)=\frac{c(x)}{x}[/tex]
Now we get [tex]f(x)= 7x^2-70x+13000[/tex]
f(x) is continuously differentiable for all x
Here x≥0 since it represents the number of items.,
Put x=0 in [tex]7x^2-70x+13000[/tex]
For x=0 the average cost becomes 13000
[tex]f(0)=7(0)^2-70(0)+13000[/tex]
[tex]=13000[/tex]
∴ f(0)=13000
To find Local extrema :
Differentiating f(x) with respect to x
[tex]f^{\prime} (x)=14x-70=0[/tex]
[tex]14x=70[/tex]
[tex]x=\frac{70}{14}[/tex]
∴ x=5 gives the minimum average cost .
At x=5 the average cost is
[tex]f(5)=7(5)^2-70(5)+13000[/tex]
[tex]=12825[/tex]
