The profit (in thousands of dollars) of a company is represented as P=−5x2+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 toward marketing to achieve the maximum profit?

Answer 1

To determine the amount spent on marketing that will achieve the maximum profit, we analyze the profit function given by:

$$P=-5x^2+1000x+5000$$

This is a quadratic equation in the standard form $P=a{x}^2+bx+c$, where $a=-5$, $b=1000$, and $c=5000$. Since $a<0$, the parabola opens downwards and has a maximum point at its vertex.

The x-coordinate of the vertex of a quadratic function can be calculated using the formula:

$$x=-\frac{b}{2a}$$

Substituting the values of $a$ and $b$:

$$x=-\frac{1000}{2\times -5}=-\frac{1000}{-10}=100$$

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

$$\boxed{{\displaystyle 100}}\text{ (in thousands of dollars)}$$