Suppose a firm's total revenues depend on the amount produced ( \(q\) ) according to the function \\[ R=70 q-q^{2} \\] Total costs also depend on \(q\) \\[ C=q^{2}+30 q+100 \\] a. What level of output should the firm produce to maximize profits \((R-C) ?\) What will profits be? b. Show that the second-order conditions for a maximum are satisfied at the output level found in part (a). c. Does the solution calculated here obey the "marginal revenue equals marginal cost" rule? Explain.

First, we need to find the profit function, which is the difference between total revenues and total costs. The profit function is given by: \\[ P(q) = R(q) - C(q) = (70q - q^2) - (q^2 + 30q + 100) \\] Simplify the profit function: \\[ P(q) = 70q - q^2 - q^2 - 30q - 100 = -2q^2 + 40q - 100 \\]

In order to find the maximum profit, we should find the first derivative of the profit function with respect to \(q\) and then set it to zero. The first derivative is given by: \\[ \frac{dP}{dq} = \frac{d}{dq}(-2q^2 + 40q - 100) \\] Differentiate with respect to \(q\): \\[ \frac{dP}{dq} = -4q + 40 \\]

Now we'll set the first derivative equal to zero and solve for \(q\): \\[ -4q + 40 = 0 \Rightarrow q = 10 \\] At \(q=10\), the profit function has a critical point. Next, let's verify that it's a maximum.

To check if the second-order conditions are satisfied at \(q=10\), find the second derivative of the profit function: \\[ \frac{d^2 P}{dq^2} = \frac{d^2}{dq^2}(-2q^2 + 40q - 100) \\] Differentiate with respect to \(q\) again: \\[ \frac{d^2 P}{dq^2} = -4 \\]

Since the second derivative of the profit function is negative, i.e., \(\frac{d^2 P}{dq^2} = -4 < 0\), we conclude that the critical point at \(q=10\) represents a maximum. Now, we'll find the maximum profit value by substituting \(q=10\) into the profit function: \\[ P(10) = -2(10)^2 + 40(10) - 100 = 100 \\] Hence, the firm should produce 10 units of output to maximize its profit at $100.

Now let's check whether the marginal revenue (\(MR\)) equals marginal cost (\(MC\)) at \(q=10\). First, find the first derivatives of the total revenue and total cost functions: \\[ \frac{dR}{dq} = \frac{d}{dq}(70q - q^2) = 70 - 2q \\ \frac{dC}{dq} = \frac{d}{dq}(q^2 + 30q + 100) = 2q + 30 \\ \\] Next, find the marginal revenue and marginal cost at \(q=10\): \\[ MR(10) = 70 - 2(10) = 50 \\ MC(10) = 2(10) + 30 = 50 \\] At \(q=10\), \(MR=MC=50\). Therefore, the solution follows the "marginal revenue equals marginal cost" rule. To summarize: a. The firm should produce \(10\) units of output to maximize its profit at $100. b. The second-order conditions for a maximum are satisfied at the output level found in part (a) since the second derivative of the profit function is negative. c. The "marginal revenue equals marginal cost" rule is satisfied at the calculated output level (\(q=10\)), as the marginal revenue and marginal cost are equal at this point.