To analyze the monthly profit function given by the smoothie-making company, we will examine the provided function and its alternative forms. The original function is given by [tex]\( f(x) = 14x - 0.01x^2 - 2600 \)[/tex].
### Step 1: Standard Form
The function [tex]\( f(x) \)[/tex] is a quadratic function in the standard form [tex]\( ax^2 + bx + c \)[/tex]:
[tex]\[ f(x) = -0.01x^2 + 14x - 2600 \][/tex]
### Step 2: Vertex Form
The vertex form of a quadratic function is useful for determining the maximum or minimum profit the company can achieve:
[tex]\[ f(x) = -0.01(x - 750)^2 + 3025 \][/tex]
From this form, we can determine the following:
- The vertex of the parabola is at [tex]\( x = 750 \)[/tex].
- The maximum profit occurs at the vertex because the coefficient of [tex]\( (x-750)^2 \)[/tex] is negative, indicating a downward-opening parabola.
- The maximum profit is [tex]\( f(750) = 3025 \)[/tex] dollars.
### Step 3: Factored Form
The function can also be written in its factored form:
[tex]\[ f(x) = -0.01(x - 200)(x - 1300) \][/tex]
From this form, we can determine the following:
- The roots of the quadratic equation are [tex]\( x = 200 \)[/tex] and [tex]\( x = 1300 \)[/tex]. These are the points where the profit is zero.
- The point halfway between the roots, [tex]\( x = \frac{200 + 1300}{2} = 750 \)[/tex], confirms the vertex found in the vertex form.
### Step 4: Analysis and Conclusion
To summarize, the company's profit function [tex]\( f(x) \)[/tex] shows that:
- The company has zero profit at [tex]\( x = 200 \)[/tex] and [tex]\( x = 1300 \)[/tex] units sold.
- The maximum profit occurs at [tex]\( x = 750 \)[/tex] units sold, and the maximum profit amount is 3025 dollars.
- The profit function is quadratic, with a downward opening (as indicated by the negative coefficient of [tex]\( x^2 \)[/tex]).
Based on the given function and its alternative forms, the detailed analysis concludes that the maximum monthly profit the smoothie-making company can achieve is 3025 dollars, occurring at 750 units sold.
