P = -5x^2 + 1000x + 5000, 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 toward marketing to achieve the maximum profit

Answer 1

To find the amount spent on marketing that will achieve the maximum profit, we can analyze the quadratic function given, which is:

\[ P = -5x^2 + 1000x + 5000 \]

A quadratic function of the form \( ax^2 + bx + c \) has its maximum (if \( a < 0 \)) at the vertex, which can be found using the vertex formula:

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

In our case:

• \( a = -5 \)
• \( b = 1000 \)

Substituting these values into the formula:

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

Therefore, to achieve the maximum profit, \( x \) should be 100 (thousands of dollars).

So, the amount that should be spent on marketing is 100 thousand dollars.