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

Answer 1

To find the amount spent on marketing that will yield the maximum profit, we need to find the vertex of the quadratic equation given by:

\[ P = -5x^2 + 1,000x + 5,000 \]

In a quadratic equation of the form \( ax^2 + bx + c \), the x-coordinate of the vertex (which gives the maximum or minimum value) is found using the formula:

\[ x = -\frac{b}{2a} \]

In our equation:

• \( a = -5 \)
• \( b = 1,000 \)

Substituting these values into the formula, we get:

\[ x = -\frac{1,000}{2 \times -5} = -\frac{1,000}{-10} = 100 \]

Therefore, the amount spent on marketing to achieve the maximum profit is:

\[ \boxed{100} \text{ (in thousands of dollars)} \]

This means the company should spend $100,000 on marketing to achieve maximum profit.