The marginal cost of a product can be thought of as cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is $6.20, it cost $6.20 to increase production from 49 units of output. Suppose the marginal cost C (in dollars) to produce X thousand MP3 players is given by the function c(x)=x^2-100x+7400.A.) how many players should be produced to minimize the marginal cost?B.) what is the minimum marginal cost?

In order to find the number of players that should be produced to minimize the marginal cost, we just need to calculate the x-coordinate of the quadratic equation vertex.

This vertex represents the minimum point of the function (the point with the smallest value of y).

So, to find the vertex x-coordinate, we can use the following formula, after comparing the function with the standard form:

[tex]\begin{gathered} f(x)=ax^2+bx+c \\ c(x)=x^2-100x+7400 \\ a=1,b=-100,c=7400 \\ \\ x_v=\frac{-b}{2a}=\frac{-\mleft(-100\mright)}{2}=\frac{100}{2}=50 \end{gathered}[/tex]

Therefore 50 players should be produced to minimize the marginal cost.

b)

To calculate the minimum marginal cost, let's use the number of players found in item a in the equation for the marginal cost:

[tex]\begin{gathered} c(x)=x^2-100x+7400 \\ c(50)=50^2-100\cdot50+7400 \\ c(50)=2500-5000+7400 \\ c(50)=4900 \end{gathered}[/tex]

Therefore the minimum marginal cost is $4,900.