The profit (in thousands of dollars) of a company is given by. P = -5x^2 + 1,000x + 5,000, where P represents the profit and x represents the amount spent on marketing (in thousands of dollars). How much spending in the thousands will be directed towards marketing to achieve the maximum profit?

Answer 1

To find the amount of spending on marketing that will achieve the maximum profit for the company given the profit function \( P = -5x^2 + 1000x + 5000 \), we need to determine the vertex of the quadratic function. The vertex form of a parabola represented by \( y = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).

In this case:

• \( a = -5 \)
• \( b = 1000 \)

Using the vertex formula:

\[ x = -\frac{b}{2a} = -\frac{1000}{2(-5)} = -\frac{1000}{-10} = 100 \]

Thus, the amount of spending in thousands that will be directed towards marketing to achieve the maximum profit is \( \boxed{100} \).