The profit function is given as:

$P(x) = -0.002x^2 + 3.5x - 1300$

This is a quadratic function of the form:

$P(x) = ax^2 + bx + c$

Where:

• $a = -0.002$
• $b = 3.5$
• $c = -1300$

Since $a < 0$, the parabola opens downwards, meaning there is a maximum point at the vertex of the parabola.

The formula for the vertex $x$ of a quadratic function $P(x) = ax^2 + bx + c$ is:

$x = \frac{-b}{2a}$

Substitute the values for $a$ and $b$:

$x = \frac{-3.5}{2(-0.002)} = \frac{-3.5}{-0.004} = 875$

Therefore, the company must sell 875 patterns each month to achieve the maximum profit.

Would you like more details on this solution?

Here are five related questions:

1. How can you calculate the maximum profit using this value of $x$?
2. What if the quadratic had a positive leading coefficient? How would it affect the result?
3. How do you find the vertex of a parabola geometrically?
4. How do we interpret the value of $b$ in real-world problems?
5. What would happen to the profit function if fixed costs increased?

Tip: Always verify if a quadratic function opens upwards or downwards by checking the sign of the leading coefficient $a$.