1.Given the marginal cost function of a producer as MC=3Q2+4Q-4. Find the output level that minimizes the cost of the producer and sketch the total cost curve and the marginal cost curve of the producer.
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To find the output level that minimizes the cost of the producer, we need to find the output level at which the marginal cost is equal to zero. In other words, we want to find the point on the marginal cost curve where it intersects the x-axis.
Given the marginal cost function MC = 3Q^2 + 4Q - 4, we can set MC equal to zero and solve for Q:
3Q^2 + 4Q - 4 = 0
One way to solve this quadratic equation is by factoring or using the quadratic formula. However, factoring won't work in this case since the equation does not factor easily. So let's use the quadratic formula:
Q = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 3, b = 4, and c = -4:
Q = ( -4 ± √(4^2 - 4(3)(-4)) ) / (2(3))
= ( -4 ± √(16 + 48) ) / 6
= ( -4 ± √(64) ) / 6
= ( -4 ± 8 ) / 6
So we have two possible values for Q:
Q1 = (-4 + 8) / 6 = 4/6 = 2/3
Q2 = (-4 - 8) / 6 = -12/6 = -2
Since output level cannot be negative, we discard Q2 = -2 as the solution. Therefore, the output level that minimizes the cost of the producer is Q = 2/3.
Now, let's sketch the total cost curve and the marginal cost curve of the producer:
To sketch the total cost curve, we need the total cost function. Since the marginal cost function represents the rate at which total cost is changing, we can integrate the marginal cost function to obtain the total cost function.
To get the total cost function, let's integrate the marginal cost function MC = 3Q^2 + 4Q - 4:
TC = ∫(3Q^2 + 4Q - 4)dQ
Integrating term by term, we get:
TC = Q^3 + 2Q^2 - 4Q + C
Here, C is the constant of integration. To determine the value of C, we need additional information, such as the total cost at a given point.
Now, let's sketch the marginal cost curve. The marginal cost curve shows how the cost of producing one additional unit changes as output increases. We can plot some points on the marginal cost curve by substituting different values of Q into the marginal cost function and calculating the corresponding values of MC.
For example, let's calculate MC when Q = 0, Q = 1, and Q = 2:
When Q = 0:
MC = 3(0)^2 + 4(0) - 4 = -4
When Q = 1:
MC = 3(1)^2 + 4(1) - 4 = 3 + 4 - 4 = 3
When Q = 2:
MC = 3(2)^2 + 4(2) - 4 = 12 + 8 - 4 = 16
Now, plot these points on a graph with Q on the x-axis and MC on the y-axis. Connect the points with a smooth curve to complete the sketch of the marginal cost curve.
Note that without a specific value for C, we cannot sketch the exact shape of the total cost curve. However, the shape of the total cost curve will typically resemble the shape of the marginal cost curve. So you can use the general shape of the marginal cost curve as a guide to sketch the total cost curve.