Answer
$C(x)=5x-\displaystyle \frac{x^{2}}{20,000}+20,000.$
Work Step by Step
The marginal cost at value x is
$C'(x)=5-\displaystyle \frac{x}{10,000}$
So, $C(x)$ is an antiderivative:
$C(x)=\displaystyle \int(5-\frac{x}{10,000})dx$
$=5x-\displaystyle \frac{1}{10,000}\cdot\frac{x^{2}}{2}+D$
$=5x-\displaystyle \frac{x^{2}}{20,000}+D$
The indefinite integral gives us a collection of functions.
To find the exact function, we find $D.$
We are given, indirectly, through the text:
$C(0)=20,000\qquad$(fixed cost = cost regardless of production), so
$20,000=0-0+D$
$D=20,000$
Thus,
$C(x)=5x-\displaystyle \frac{x^{2}}{20,000}+20,000.$
