Question

find the cost function if the marginal cost function is c(x) = 9x - 2 and the fixed cost is $6.
c(x) = \square

Explanation:

Step1: Integrate marginal cost

To find the cost function \( C(x) \), we integrate the marginal cost function \( C'(x) = 9x - 2 \). The integral of \( 9x - 2 \) with respect to \( x \) is \( \int (9x - 2) dx \). Using the power rule for integration, \( \int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (where \( n
eq - 1\)), we have:
\( \int(9x - 2)dx=9\times\frac{x^{2}}{2}-2x + K=\frac{9}{2}x^{2}-2x + K\), where \( K \) is the constant of integration.

Step2: Determine the constant \( K \)

We know that the fixed cost is the cost when \( x = 0 \), i.e., \( C(0)=6 \). Substitute \( x = 0 \) and \( C(0) = 6 \) into the function \( C(x)=\frac{9}{2}x^{2}-2x + K \):
\( C(0)=\frac{9}{2}(0)^{2}-2(0)+K=K \)
Since \( C(0) = 6 \), then \( K = 6 \).

Step3: Write the cost function

Substitute \( K = 6 \) back into the integrated function:
\( C(x)=\frac{9}{2}x^{2}-2x + 6 \)

Answer:

\( \frac{9}{2}x^{2}-2x + 6 \)