To determine the correct profit function [tex]\(P(x)\)[/tex] for the magazine company, let's go through the step-by-step process of using the given cost function [tex]\(C(x)\)[/tex] and revenue function [tex]\(R(x)\)[/tex].
### Step-by-Step Solution:
1. Identify the given functions:
- Cost function [tex]\(C(x) = 700x^2 + 400x\)[/tex]
- Revenue function [tex]\(R(x) = -0.4x^3 + 900x^2 - 200x + 600\)[/tex]
2. Formulate the profit function [tex]\(P(x)\)[/tex]:
The profit function [tex]\(P(x)\)[/tex] is given by the difference between revenue and cost:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
3. Substitute [tex]\(R(x)\)[/tex] and [tex]\(C(x)\)[/tex] into [tex]\(P(x)\)[/tex]:
[tex]\[
P(x) = \left(-0.4x^3 + 900x^2 - 200x + 600\right) - \left(700x^2 + 400x\right)
\][/tex]
4. Simplify the expression:
[tex]\[
P(x) = -0.4x^3 + 900x^2 - 200x + 600 - 700x^2 - 400x
\][/tex]
Combine like terms:
[tex]\[
P(x) = -0.4x^3 + (900x^2 - 700x^2) + (-200x - 400x) + 600
\][/tex]
[tex]\[
P(x) = -0.4x^3 + 200x^2 - 600x + 600
\][/tex]
5. Compare with the given choices:
- Option A: [tex]\(P(x) = -0.4x^3 + 200x^2 - 600x + 600\)[/tex]
- Option B: [tex]\(P(x) = 0.4x^3 + 200x^2 - 600x + 600\)[/tex]
6. Determine the correct option:
- The simplified profit function is [tex]\(P(x) = -0.4x^3 + 200x^2 - 600x + 600\)[/tex], which matches Option A.
Therefore, the correct profit function is:
[tex]\[
\boxed{-0.4x^3 + 200x^2 - 600x + 600}
\][/tex]
So, the correct answer is Option A.
