C'(x)=
50
_______________
square root of x
fixed cost= $25000
find the cost function C(x)
integrate C'(x)
wouldn't that be 100x^(1/2) + c ???
now where do you think the fixed cost comes in?
To find the cost function C(x) from the given derivative C'(x) = 50 / sqrt(x), you need to integrate C'(x) with respect to x.
Integrating a function involves finding the antiderivative of that function. In this case, to integrate C'(x), you can treat 50 / sqrt(x) as 50x^(-1/2) and then use the power rule of integration:
∫50x^(-1/2) dx = 50 * (x^(1/2)) / (1/2) + C
= 100 * sqrt(x) + C
So, you are correct that the antiderivative of C'(x) would be 100 * sqrt(x) + C, where C represents the constant of integration.
Now, let's consider the fixed cost. The fixed cost represents the cost incurred regardless of the quantity produced or sold. In this case, the fixed cost is given as $25,000.
Since the fixed cost does not vary with the quantity of production, it can be considered as the value of C. So, we have:
C(x) = 100 * sqrt(x) + C
Substituting the fixed cost C = $25,000:
C(x) = 100 * sqrt(x) + $25,000
Therefore, the cost function C(x) is given by 100 * sqrt(x) + $25,000.