Suppose TU = 2Q^3-3Q^2+8Q+20 then find the level of output where ATC is minimum?
To find the level of output where Average Total Cost (ATC) is minimum, we need to first find the formula for ATC using the Total Cost (TC) and the level of output (Q).
ATC is given by the formula: ATC = TC / Q
To find TC, we have the formula: TC = TU - TVC, where TU is Total Utility and TVC is Total Variable Cost.
In this case, we are given the formula for TU: TU = 2Q^3 - 3Q^2 + 8Q + 20
Now, we need to find the formula for TVC. TVC is equal to Total Cost (TC) minus Total Fixed Cost (TFC).
Since we are not given the TFC, we will assume it to be a constant value, which will cancel out when finding the minimum value of ATC.
Therefore, we can write TVC as TVC = TC - TFC = TC.
Now, we can substitute the formula for TC in the formula for TVC:
TVC = TU - TVC
Simplifying the equation, we get:
2Q^3 - 3Q^2 + 8Q + 20 = TVC
Next, we substitute the value of TVC in the ATC formula:
ATC = TC / Q = (2Q^3 - 3Q^2 + 8Q + 20) / Q
Simplifying the equation, we get:
ATC = 2Q^2 - 3Q + 8 + 20/Q
To find the level of output where ATC is minimum, we need to take the derivative of ATC with respect to Q and set it equal to zero.
d(ATC)/dQ = 4Q - 3 - 20/Q^2 = 0
Simplifying the equation, we get:
4Q - 3 - 20/Q^2 = 0
Multiplying by Q^2 to get rid of the fraction, we get:
4Q^3 - 3Q^2 - 20 = 0
Now, to solve this equation for Q, you would need to use numerical methods or calculators to find the values of Q that satisfy the equation.