The question is incomplete. The complete question is :
A manufacturer believes that the cost function : [tex]$C(x) =\frac{5}{2}x^2+120 x+560$[/tex] approximates the dollar cost of producing x units of a product. The manu- facturer believes it cannot make a profit when the marginal cost goes beyond $210. What is the most units the manufacturer can produce and still make a profit? What is the total cost at this level of production?
Solution :
Given the cost function is :
[tex]$C(x) =\frac{5}{2}x^2+120 x+560$[/tex]
Now, Marginal cost = [tex]$\frac{d}{dx}C(x)$[/tex]
So, if the marginal cost = $ 210, then the manufacturer also makes a profit and if it goes beyond $ 210 than the manufacturer cannot make a profit.
Therefore, we have to equate : [tex]$\frac{d}{dx}C(x)= \$ 210$[/tex]
[tex]$\frac{d}{dx}C(x)= \frac{5}{2}(2x)+120 = 210$[/tex]
[tex]$5x + 120 = 210$[/tex]
[tex]$5x=210-120$[/tex]
[tex]$5x=90$[/tex]
[tex]$x=45$[/tex]
So when x = 45, then C(x) = $ 8042.5
Therefore, the manufacturer [tex]$\text{can make up}$[/tex] to 45 units and [tex]$\text{still makes a profit.}$[/tex] This leads to a total cost of $ 8042.5
