Q5 The profit (in thousands of dollars) of a tech company is represented asP = -4x2 + 800x + 3,000, 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 find the amount spent on marketing that will achieve the maximum profit for the given profit function \( P = -4x^2 + 800x + 3000 \), we can use the vertex formula for a quadratic function. The vertex \( x \) of a parabola represented in the standard form \( P = ax^2 + bx + c \) occurs at the value:

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

In this case, the coefficients are:

• \( a = -4 \)
• \( b = 800 \)

Now, substituting these values into the vertex formula:

\[ x = -\frac{800}{2 \times -4} = -\frac{800}{-8} = 100 \]

Therefore, the amount directed toward marketing to achieve the maximum profit is \( x = 100 \) (in thousands of dollars).

Thus, the spending directed toward marketing to achieve the maximum profit is $100,000.