The profit (in thousands of dollars) of a company is represented as P=−5x2+1,000x+5,000%0D%0A%0D%0A=%0D%0A−%0D%0A5%0D%0A%0D%0A2%0D%0A+%0D%0A1%0D%0A,%0D%0A000%0D%0A%0D%0A+%0D%0A5%0D%0A,%0D%0A000%0D%0A, where P%0D%0A%0D%0A 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?
To find the amount of spending on marketing that will achieve the maximum profit, we need to determine the value of x that corresponds to the maximum value of the profit function P.
The profit function is given by P = -5x^2 + 1,000x + 5,000.
This is a quadratic equation in the form of ax^2 + bx + c, where a = -5, b = 1,000, and c = 5,000.
The x-coordinate of the vertex of a quadratic function in standard form, ax^2 + bx + c, is given by x = -b/2a.
In this case, x = -1,000/(2*-5) = -1,000/-10 = 100.
Therefore, spending $100,000 in marketing (in thousands of dollars) will achieve the maximum profit.