a. To find the value of L that maximizes output, we need to take the derivative of the production function with respect to L and set it equal to zero, and then solve for L.
Q = 6L^2 - 0.4L^3
Taking the derivative with respect to L:
dQ/dL = 12L - 1.2L^2
Setting it equal to zero:
12L - 1.2L^2 = 0
Dividing both sides by L:
12 - 1.2L = 0
1.2L = 12
L = 10
Therefore, the value of L that maximizes output is 10.
b. Marginal product is the derivative of the production function with respect to L:
MP = dQ/dL = 12L - 1.2L^2
To find the value of L that maximizes marginal product, we need to set the derivative equal to zero and solve for L:
12L - 1.2L^2 = 0
Dividing both sides by L:
12 - 1.2L = 0
1.2L = 12
L = 10
Therefore, the value of L that maximizes marginal product is 10.
c. Average product is given by AP = Q/L. To find the value of L that maximizes average product, we need to find the value of L that maximizes Q/L.
AP = Q/L = (6L^2 - 0.4L^3)/L = 6L - 0.4L^2
To maximize AP, we need to take the derivative of AP with respect to L and set it equal to zero:
d(AP)/dL = 6 - 0.8L = 0
0.8L = 6
L = 7.5
Therefore, the value of L that maximizes average product is 7.5.
Consider the following short run production function : Q=6L2-0.4L3
a. Find the value of L that maximizes out put
b. Find the value of L that maximizes marginal product
c. Find the value of L that maximizes average product
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