The problem in the image provides two functions: Total Revenue (TR) and Total Cost (TC), and asks for conditions that maximize profit.

We are given:

• $TR = 300q - 22q^2$
• $TC = 162q^2 - 99q^3 + 60q + 30$

To maximize profit, we follow these steps:

1. Profit Function: Profit ($\pi$) is the difference between total revenue (TR) and total cost (TC). $\pi(q) = TR(q) - TC(q)$ Substituting the given functions: $\pi(q) = (300q - 22q^2) - (162q^2 - 99q^3 + 60q + 30)$ Simplify the profit function.

2. First Derivative (Marginal Profit): To find the quantity $q$ that maximizes profit, take the derivative of the profit function with respect to $q$ and set it equal to zero.

3. Second Derivative: Use the second derivative to check if the critical point found in step 2 is a maximum.

I'll now calculate and simplify the profit function before moving on to the next steps.The profit function is:

$\pi(q) = 99q^3 - 184q^2 + 240q - 30$

Next, I'll find the first derivative of this function (Marginal Profit) to determine the critical points, which will help us find the quantity $q$ that maximizes profit.It seems like I can't do more advanced data analysis right now. Please try again later.

For now, here's a manual outline of the next steps you can follow:

1. Take the derivative of the profit function: $\frac{d\pi(q)}{dq} = 3 \times 99q^2 - 2 \times 184q + 240$ This simplifies to: $297q^2 - 368q + 240$

2. Set this first derivative equal to zero and solve for $q$ to find the critical points.

3. Finally, use the second derivative test to confirm if the critical points represent a maximum.

If you'd like help with solving this or have questions, feel free to ask!

Here are 5 related questions:

1. How do you solve for critical points using derivatives?
2. What is the significance of the second derivative in optimization problems?
3. How does marginal cost and marginal revenue relate to profit maximization?
4. What happens if there are multiple critical points in a profit function?
5. How do changes in costs affect profit maximization?

Tip: In optimization problems, always verify if critical points are maxima or minima using the second derivative.