P(q1,q2)=(70-.05(q1+q2))(q1+q2)-(9.5+.04(q1)^2)-(4+.05(q2)^2) maximize this function

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).

Maximize the function P(q1, q2) = (70 - 0.05(q1 + q2))(q1 + q2) - (9.5 + 0.04q1^2) - (4 + 0.05q2^2) by finding the critical points using first-order partial derivatives and verifying the maximum with second-order conditions. A detailed step-by-step explanation suitable for undergraduate calculus students.

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