To determine the price that maximizes weekly profit, you need to analyze the relationship between price, demand, and cost. Here's a general step-by-step method:

1. Define Variables:

• Let $p$ represent the price per unit of a product.
• Let $q(p)$ be the demand function, which shows the number of units sold depending on the price $p$. This is often a decreasing function of $p$, meaning higher prices reduce demand.
• Let $C(q)$ represent the cost function, which describes the total cost to produce $q$ units.

2. Profit Function:

The profit function $\Pi(p)$ is defined as: $\Pi(p) = \text{Revenue}(p) - \text{Cost}(q)$ $\Pi(p) = p \cdot q(p) - C(q(p))$ Here:

• $p \cdot q(p)$ is the revenue (price multiplied by quantity sold).
• $C(q(p))$ is the total cost of producing $q(p)$ units at that price.

3. Maximizing Profit:

To maximize profit, take the derivative of the profit function with respect to $p$, set it equal to zero, and solve for $p$. This gives you the critical points: $\frac{d\Pi(p)}{dp} = 0$

• This derivative combines the marginal revenue (how revenue changes with price) and marginal cost (how cost changes with production).

• After finding the critical points, use the second derivative test to ensure that the value of $p$ provides a maximum (not a minimum or inflection point).

4. Analyze the Result:

The value of $p$ that maximizes the profit will depend on the specific forms of $q(p)$ and $C(q)$. For instance, if you have linear demand and constant marginal cost, the solution can be straightforward. Otherwise, the functions might be more complex, requiring further analysis.

If you provide a specific demand function or cost structure, I can walk you through solving it in more detail!

Would you like to go through an example or clarify your particular scenario?

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Here are 5 related questions:

1. What is your current demand function for the product, if known?
2. Do you have data on how price affects the number of units sold?
3. What are your production costs (fixed and variable)?
4. Are there any constraints on how low or high the price can go?
5. Have you considered competitor pricing when deciding on a price point?

Tip: Understanding the elasticity of demand is crucial for pricing strategy—it shows how sensitive customers are to price changes!