A local sneaker company’s profits π each unit of x sneaker sold has been estimated as 
π(x)= 〖-x〗^2+300x-9000
	a)Find the number of sneaker sold in order for the company to maximize its profit. 
	b)What is the maximum level of profit can the company achieve?
	c)Find the number of sneaker that the company have to sell in order to breakeven. 
	d)Sketch the graph for the profit function for this company. 



Let's analyze the given profit function for the local sneaker company again to ensure we cover all parts clearly: 
\[ \pi(x) = -x^2 + 300x - 9000 \]

### Part (a): Find the number of sneakers sold to maximize profit

To maximize the profit, we need to find the vertex of the parabola represented by the quadratic function. For a quadratic function in the form $ \pi(x) = ax^2 + bx + c $, the x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]

Here, $a = -1$, $b = 300$, and $c = -9000$. Plugging in these values:
\[ x = -\frac{300}{2(-1)} = \frac{300}{2} = 150 \]

So, the number of sneakers sold to maximize profit is **150**.

### Part (b): What is the maximum level of profit the company can achieve?

To find the maximum profit, substitute $ x = 150 $ back into the profit function $ \pi(x) $:
\[ \pi(150) = -(150)^2 + 300(150) - 9000 \]
\[ \pi(150) = -22500 + 45000 - 9000 \]
\[ \pi(150) = 13500 \]

So, the maximum level of profit is **$13,500**.

### Part (c): Find the number of sneakers that the company has to sell in order to breakeven

To find the breakeven points, solve for $ x $ when $ \pi(x) = 0 $:
\[ -x^2 + 300x - 9000 = 0 \]

This quadratic equation can be solved using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Plugging in $ a = -1 $, $ b = 300 $, and $ c = -9000 $:
\[ x = \frac{-300 \pm \sqrt{300^2 - 4(-1)(-9000)}}{2(-1)} \]
\[ x = \frac{-300 \pm \sqrt{90000 - 36000}}{-2} \]
\[ x = \frac{-300 \pm \sqrt{54000}}{-2} \]
\[ x = \frac{-300 \pm 232.38}{-2} \]

This gives us two solutions:
\[ x = \frac{-300 + 232.38}{-2} \approx 33.81 \]
\[ x = \frac{-300 - 232.38}{-2} \approx 266.19 \]

So, the company will breakeven when they sell approximately **34 or 266 sneakers**.

### Part (d): Sketch the graph for the profit function

Here is the graph again for the profit function $\pi(x) = -x^2 + 300x - 9000$:

- The parabola opens downward.
- The vertex, indicating the maximum profit, is at $ (150, 13500) $.
- The breakeven points are approximately at $ x = 34 $ and $ x = 266 $.

![Profit Function Graph](sandbox:/mnt/data/2024-07-16/profit_function_graph.png)

Do you want details or have any questions?

Here are 5 questions you can ask next:
1. How can I find the derivative of a function?
2. What is the significance of the vertex in a quadratic function?
3. Can you explain the quadratic formula in detail?
4. How do you determine if a function is increasing or decreasing?
5. What are some applications of quadratic functions in real life?

Tip: When working with quadratic functions, identifying the vertex and the direction in which the parabola opens can help you understand the function's maximum or minimum values and overall behavior.

Explore how to maximize profit and determine breakeven points using a quadratic function. Learn the steps to find the number of sneakers to sell for maximum profit and breakeven, with detailed calculations and graphical analysis.

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