The image provides a revenue function: \[ R(q_A, q_B) = 60q_A + 80q_B - 6q_A^2 - 9q_B^2 - 4q_Aq_B \] The task is to **maximize the revenue** by choosing appropriate values for \(q_A\) and \(q_B\). ### To maximize the revenue: 1. **Take partial derivatives** of \(R\) with respect to \(q_A\) and \(q_B\). 2. **Set the derivatives to zero** to find critical points. 3. **Solve the resulting system of equations** for \(q_A\) and \(q_B\). Would you like to proceed step by step with the calculations?

Certainly! Let's proceed step by step to maximize the revenue function: --- ### **Step 1: Find the critical points by taking partial derivatives** Given the revenue function: \[ R(q_A, q_B) = 60q_A + 80q_B - 6q_A^2 - 9q_B^2 - 4q_Aq_B \] **Partial derivatives:** - With respect to \(q_A\): \[ \frac{\partial R}{\partial q_A} = 60 - 12q_A - 4q_B \] - With respect to \(q_B\): \[ \frac{\partial R}{\partial q_B} = 80 - 18q_B - 4q_A \] **Explanation:** We differentiate each term with respect to the variables. Constants are eliminated, and the powers of the variables are reduced by one, applying the power rule. The mixed term \(-4q_Aq_B\) contributes \(-4q_B\) when differentiating w.r.t. \(q_A\), and \(-4q_A\) when differentiating w.r.t. \(q_B\). --- ### **Step 2: Set the derivatives to zero and solve for \(q_A\) and \(q_B\)** Set the equations: \[ 60 - 12q_A - 4q_B = 0 \quad \Rightarrow \quad 12q_A + 4q_B = 60 \quad (1) \] \[ 80 - 18q_B - 4q_A = 0 \quad \Rightarrow \quad 4q_A + 18q_B = 80 \quad (2) \] **Now, solve this system:** Multiply equation (1) by 4: \[ (1) \times 4: \quad 48q_A + 16q_B = 240 \] Rewrite equation (2): \[ 4q_A + 18q_B = 80 \] Now, subtract the second from the first: \[ (48q_A + 16q_B) - (4q_A + 18q_B) = 240 - 80 \] \[ (48q_A - 4q_A) + (16q_B - 18q_B) = 160 \] \[ 44q_A - 2q_B = 160 \] Solve for \(q_A\): \[ 44q_A = 160 + 2q_B \quad \Rightarrow \quad q_A = \frac{160 + 2q_B}{44} = \frac{80 + q_B}{22} \] Substitute \(q_A\) back into equation (1): \[ 12 \times \frac{80 + q_B}{22} + 4q_B = 60 \] Multiply through by 22 to clear denominator: \[ 12(80 + q_B) + 22 \times 4q_B = 60 \times 22 \] \[ 12 \times 80 + 12q_B + 88q_B = 1320 \] \[ 960 + 12q_B + 88q_B = 1320 \] Combine like terms: \[ 960 + 100q_B = 1320 \] Subtract 960: \[ 100q_B = 360 \] Solve for \(q_B\): \[ q_B = \frac{360}{100} = 3.6 \] Now, find \(q_A\): \[ q_A = \frac{80 + 3.6}{22} = \frac{83.6}{22} \approx 3.8 \] --- ### **Final answer:** \[ \boxed{ q_A \approx 3.8,\quad q_B \approx 3.6 } \] These values maximize the revenue based on the given function. --- Would you like to verify if this critical point corresponds to a maximum?

Actually, based on our calculations, the approximate optimal values are: \[ q_A \approx 3.8 \quad \text{and} \quad q_B \approx 3.6 \] So, **you should write:** \[ q_A \approx 3.8,\quad q_B \approx 3.6 \] to represent the values that maximize the revenue. **Note:** If you prefer to keep the numbers consistent with your calculations or rounding preferences, ensure your final answer reflects the approximations clearly.