When does a company get the maximum profit if: Marginal Cost = dC/dq = 6q + 20 and P = 100-q, where q = output. Fixed costs are 600.

From the information provided I assume you want the number
of units produced to maximise profit.

The marginal cost is
given as dC/dq = 6q + 20

Total cost is the integral of the
marginal cost or C = Int[6q + 20] = 6q^2/2 + 20q + F = 3q^2 + 20q +
F

As the fixed costs are 600, F = 600 in the function
obtained earlier.

The total cost for q units is 3q^2 + 20q
+ 600.

I assume the revenue per unit is 100 - q. The
revenue when q units are produced 100q - q^2.

Total profit
for q units is TP = Revenue - Total cost

=> 100q -
q^2 - (3q^2 + 20q + 600)

=> 100q - q^2 - 3q^2 - 20q
- 600

=> -4q^2 + 80q -
600

For profit maximization we need to differentiate -4q^2
+ 80q - 600 with respect to q and solve for q.

d (TP) / dq
= -8q + 80

-8q + 80 =
0

=> q =
10

The company makes the maximum profit when
it manufactures 10 units.