Final answer:
To find the firm's profit function, subtract the total cost function from the total revenue function. Then, find the derivative of the profit function to get the marginal profit function. Substitute the given demand value to compute the marginal profit. Analyze the sign of the marginal profit to determine if profit is increasing or decreasing at a specific demand level. Find the level of demand that maximizes profit by setting the marginal profit equal to zero and solve for the quantity. Finally, calculate the amount of profit obtained at the maximum by plugging in the value of quantity into the profit function.
Step-by-step explanation:
In order to find the firm's profit function, we need to subtract the total cost function from the total revenue function. The total revenue function is given by price multiplied by quantity, which can be represented as TR(x) = px. Therefore, the profit function is given by P(x) = TR(x) - TC(x) = px - (0.4x^2 + 3x + 40).
To find the marginal profit, we need to find the derivative of the profit function with respect to quantity. The marginal profit function is given by MP(x) = dP(x)/dx = p - (0.8x + 3).
To compute the marginal profit when demand is x = 10, we substitute x = 10 into the marginal profit function and solve: MP(10) = p - (0.8(10) + 3).
To determine whether profit is increasing or decreasing when demand is x = 10, we need to analyze the sign of the marginal profit. If MP(10) > 0, profit is increasing, and if MP(10) < 0, profit is decreasing.
To find the level of demand that maximizes the firm's profit, we need to find the quantity where marginal profit is equal to zero. Set MP(x) = 0 and solve for x. Finally, to calculate the amount of profit obtained at the maximum, substitute the value of x into the profit function.