The given profit function is:

$P(x) = -0.75x^2 + 6x - 9$

To determine when there will be profit, we need to find when $P(x) > 0$. This means we need to solve the inequality:

$-0.75x^2 + 6x - 9 > 0$

First, let's solve the equation:

$-0.75x^2 + 6x - 9 = 0$

This will help us identify the critical points where the profit is zero (break-even points). After that, we can determine the intervals where the function is positive (profit exists).

Let's solve this quadratic equation.The solutions to the equation $-0.75x^2 + 6x - 9 = 0$ are $x = 2$ and $x = 6$.

These are the break-even points where the profit is zero. To determine when there is a profit, we need to check the sign of the profit function in the intervals:

1. $x < 2$
2. $2 < x < 6$
3. $x > 6$

By analyzing the quadratic function, we can conclude:

• The function opens downwards (since the coefficient of $x^2$ is negative).
• The profit is positive between the break-even points, i.e., for $2 < x < 6$.

Therefore, there will be profit when $x$ (the number of units) is between 2 and 6.

Would you like a more detailed explanation or further analysis?

Here are 5 related questions:

1. How do you interpret a profit function in terms of economics?
2. What is the significance of the discriminant in quadratic equations?
3. How does the vertex of a quadratic function help in analyzing profit?
4. What are the steps to analyze when a quadratic function is positive or negative?
5. How does the shape of a parabola help in determining intervals of profit?

Tip: Always plot the graph of quadratic functions to better understand the intervals where they are positive or negative!