We are tasked with maximizing the function $P(q_1, q_2)$, which can be written as:

$P(q_1, q_2) = (70 - 0.05(q_1 + q_2))(q_1 + q_2) - \left(9.5 + 0.04q_1^2\right) - \left(4 + 0.05q_2^2\right)$

To find the maximum, we need to compute the first-order partial derivatives of $P$ with respect to $q_1$ and $q_2$, set them equal to zero, and solve the resulting system of equations. We will then check the second-order conditions to ensure we have found a maximum.

Step 1: First-order conditions

1. Partial derivative with respect to $q_1$:

$\frac{\partial P}{\partial q_1} = \frac{\partial}{\partial q_1}\left[(70 - 0.05(q_1 + q_2))(q_1 + q_2)\right] - \frac{\partial}{\partial q_1}\left(9.5 + 0.04q_1^2\right)$

2. Partial derivative with respect to $q_2$:

$\frac{\partial P}{\partial q_2} = \frac{\partial}{\partial q_2}\left[(70 - 0.05(q_1 + q_2))(q_1 + q_2)\right] - \frac{\partial}{\partial q_2}\left(4 + 0.05q_2^2\right)$

Let me calculate these derivatives and solve for the critical points.The critical point that maximizes the function occurs at:

$q_1 = 269.23, \quad q_2 = 215.38$

These values represent the optimal quantities $q_1$ and $q_2$ that maximize the given function $P(q_1, q_2)$.

Would you like to explore further details, such as verifying if this point is indeed a maximum using second-order conditions or calculating the maximum value of the function at this point?

Here are 5 related questions to expand on this:

1. What is the maximum value of $P(q_1, q_2)$ at the critical point?
2. How do second-order conditions confirm whether this is a maximum or minimum?
3. What happens if you impose constraints on $q_1$ and $q_2$?
4. Can we analyze the sensitivity of $P$ with respect to changes in $q_1$ and $q_2$?
5. What is the economic interpretation of maximizing $P(q_1, q_2)$ in this context?

Tip: In optimization problems, it is crucial to verify if the critical points are maxima or minima by checking second-order conditions (i.e., the Hessian matrix).