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

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

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

Step-by-step 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 $f(x)=\frac{c(x)}{x}$

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

$f^{\prime} (x)=14x-70=0$

$14x=70$

$x=\frac{70}{14}$

∴  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.