Question

Suppose that c (x )equals 4 x cubed minus 16 x squared plus 12 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

Therefore the number of product is 11.42 to minimize the average cost .

Explanation:

Given that,

`C(x)=4x^3-16x^2+12,000x`

where C(x) is cost of manufacturing of x items.

Differentiating with respect to x

`C'(x)=12x^2-32x+12,000`

Again differentiating with respect to x

`C''(x)=24x-32`

To find the minimum cost, we set C'(x)=0

`\therefore 12x^2-32x-1200=0`

`\Rightarrow 4(3x^2-8x-300)=0`

`\Rightarrow 3x^2-8x-300=0` [ since 4≠0]

Applying quadratic formula
`x=(-b\pm√(b^2-4ac))/(2a)`, here a= 3, b= -8, c=-300

`\therefore x=\frac {-(-8)\pm√((-8)^2-4.3.(-300))}{2.3}`

=11.42, -8.76

The number of item is negative, it can't make sense.

∴x=11.42

Now

`C''|_(x=11.42)=24(11.42)-32=242.08>0`

Therefore when x= 11.42, the cost of manufacturing will minimum.

Therefore the number of product is 11.42 to minimize the average cost .