The total cost C(x) of a firm is C(x) = 0.0005x^3 – 0.7x^2 – 30x + 3,000, where x is the output. The value of x; for which MVC = AVC is: (where VC denotes the variable cost, MVC denotes marginal variable cost and AVC denotes average variable cost
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To find the value of x for which MVC = AVC, we need to understand the definitions of marginal variable cost (MVC) and average variable cost (AVC).
MVC is the change in total cost (C) when one more unit of output (x) is produced. It is calculated by taking the derivative of the cost function with respect to x, which represents the rate of change of cost with respect to output.
AVC is the total variable cost (VC) divided by the total output (x). It represents the average cost per unit of output.
In this case, the cost function C(x) is given as C(x) = 0.0005x^3 – 0.7x^2 – 30x + 3,000.
To find the MVC, we differentiate the cost function with respect to x:
C'(x) = 0.0015x^2 - 1.4x - 30
To find the AVC, we divide the total variable cost (VC) by the total output (x).
AVC = VC / x
Now, we can set MVC equal to AVC and solve for x.
0.0015x^2 - 1.4x - 30 = VC / x
To find the value of x for which MVC = AVC, it is necessary to have information about the variable cost (VC). If that information is not provided, it is not possible to find the specific value of x.