Question

Suppose that c (x )equals 7 x cubed minus 70 x squared plus 13 comma 000 x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Answer 1

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Explanation:

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Answer 2

A production level that will minimize the average cost of making x items is x=5.

Explanation:

Given that

`c(x)=7x^3-70x^2+13,000x`

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
`f(x)=(c(x))/(x)`

Now we get
`f(x)= 7x^2-70x+13000`

f(x) is continuously differentiable for all x

Here x≥0 since it represents the number of items.,

Put x=0 in
`7x^2-70x+13000`

For x=0 the average cost becomes 13000

`f(0)=7(0)^2-70(0)+13000`

`=13000`

∴ f(0)=13000

To find Local extrema :

Differentiating f(x) with respect to x

`f^(\prime) (x)=14x-70=0`

`14x=70`

`x=(70)/(14)`

∴ x=5 gives the minimum average cost .

At x=5 the average cost is

`f(5)=7(5)^2-70(5)+13000`

`=12825`

∴ f(5)=12825 which is smaller than for x=0 is 13000

∴ f(x) is decreasing between 0 and 5 and it is increasing after 5.