Question

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?(1 point)

Answer 1

Graph the function f(x)=2x^2−2x
. What is the y-value that corresponds to the x-value of −1
?(1 point)

Answer 2

incorrect

Answer 3

To find the amount of spending on marketing that maximizes profit, we can use the formula for the vertex of a quadratic function. The profit function given is

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

This is a quadratic equation in the form \( P = ax^2 + bx + c \), where:

- \( a = -5 \)
- \( b = 1000 \)
- \( c = 5000 \)

The x-coordinate of the vertex (which represents the maximum point for a downward-opening parabola like this one) can be found using the formula:

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

Substituting the values of \( a \) and \( b \):

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

Thus, the amount of spending in thousands that will be directed toward marketing to achieve the maximum profit is:

\[
\boxed{100}
\]

Answer 4

To find the y-value that corresponds to the x-value of \(-1\) for the function \(f(x) = 2x^2 - 2x\), we need to substitute \(x = -1\) into the function.

\[
f(-1) = 2(-1)^2 - 2(-1)
\]

Calculating each term:

1. \(2(-1)^2 = 2(1) = 2\)
2. \(-2(-1) = 2\)

Now, combine the results:

\[
f(-1) = 2 + 2 = 4
\]

Thus, the y-value that corresponds to the x-value of \(-1\) is:

\[
\boxed{4}
\]