Suppose a competitive firm has as its total cost function:

$TC=21+3q^2$

Suppose the firm's output can be sold $($in integer units$)$ at $$71$ per unit.

Use calculus and formulas to find a solution $($don$\text{'}$t just build a table in a spreadsheet as in the

previous lesson$).$

Hint $1$: The first derivative of the total profit function, which is cumulative, is the marginal profit function, which is

incremental. The lecture and formula summary explain how to compute the derivative.

Set the marginal profit equal to zero to define an equation for the optimal quantity q$.$

Hint $2$: When computing the total profit for a candidate quantity, use the total profit function you define $($rather

than summing the marginal profits using the marginal profit function$).$

How many integer units should the firm produce to maximize profit?

Please specify your answer as an integer. In the case of equal profit from rounding up and down for a non$-$

integer initial solution quantity, proceed with the higher quantity.

What is the total profit at the optimal integer output level?

Please specify your answer as an integer.