To find the amount spent on marketing that will yield the maximum profit based on the given profit function, we first need to rewrite the profit equation correctly and then identify its characteristics.
The provided profit equation seems to have a minor typographical error. The correct formulation is generally a quadratic equation of the form:
\[ p = ax^2 + bx + c \]
However, it seems like the expression you've provided, \( p = -4x + 800x + 3000 \), simplifies to:
\[ p = 796x + 3000 \]
But that is a linear equation, not a quadratic one, meaning it won’t have a maximum profit; it will increase indefinitely as \( x \) increases.
If you meant a different equation and it was supposed to be a quadratic function (which might be something like \[ p = -4x^2 + 800x + 3000 \] where the coefficient of \( x^2 \) is negative), then we can find the maximum profit.
To find the vertex of a quadratic equation \( p = ax^2 + bx + c \):
- The vertex \( x \) coordinate (where maximum or minimum occurs) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Assuming your profit equation meant was \( p = -4x^2 + 800x + 3000 \):
-
Identify coefficients:
- \( a = -4 \)
- \( b = 800 \)
-
Apply the vertex formula:
\[ x = -\frac{800}{2 \cdot -4} = -\frac{800}{-8} = 100 \]
Thus, the amount of spending in thousands of dollars that should be directed toward marketing to achieve the maximum profit is 100 thousands of dollars or $100,000.
If you confirm or clarify the exact function, I can provide further details if needed!
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