Given the cost function $C(x)=3 x^{3}-4 x^{2}+13 x+6$, find the minimum marginal cost.
The minimum marginal cost is $\$$.
(Do not round until the final answer. Then round to two decimal places as needed.)
Answer
Unfortunately, this equation does not have a real solution. Therefore, the marginal cost does not have a minimum value.
Step 1 ：First, we need to find the marginal cost, which is the derivative of the cost function. So, we have \(C'(x) = 9x^{2} - 8x + 13\).
Step 2 ：Next, we need to find the minimum of this function. To do this, we set the derivative equal to zero and solve for \(x\). So, we have \(9x^{2} - 8x + 13 = 0\).
Step 3 ：Unfortunately, this equation does not have a real solution. Therefore, the marginal cost does not have a minimum value.
