Question

The marginal cost of manufacturing an item when x thousand items are produced is dC/dx= 4x^3 - 6x + 5 dolars/item.Find the cost function C(x) if C(0)=550

Answer 1

Answer:

The cost function is $C(x)=x^4-3x^2+5x+550$.

Step-by-step explanation:

It is given that the marginal cost of manufacturing an item when x thousand items are produced is

$\frac{dC}{dx}=4x^3-6x+5$

We need to find the cost function.

Multiply both sides by dx.

$dC=(4x^3-6x+5)dx$

Integrate both sides to find the cost function.

$\int dC=\int (4x^3-6x+5)dx$

$\int dC=4\int x^3dx-6\int xdx+5\int 1 dx$

$C(x)=4(\frac{x^4}{4})-6(\frac{x^2}{2})+5x+C$

where, C(x) is const function and C is a constant.

$C(x)=x^4-3x^2+5x+C$

It is given that C(0)=550. Substitute x=0 in the above function.

$C(0)=(0)^4-3(0)^2+5(0)+C$

$C(0)=C$

$550=C$

Therefore, the cost function is $C(x)=x^4-3x^2+5x+550$.