To determine the profit function, [tex]\( P(x) \)[/tex], we need to find the difference between the revenue function, [tex]\( R(x) \)[/tex], and the cost function, [tex]\( C(x) \)[/tex].
The given functions are:
- Cost function: [tex]\( C(x) = 700 x^2 + 400 x \)[/tex]
- Revenue function: [tex]\( R(x) = -0.4 x^3 + 900 x^2 - 200 x + 600 \)[/tex]
The profit function [tex]\( P(x) \)[/tex] is given by:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions into the equation for profit:
[tex]\[ P(x) = \left(-0.4 x^3 + 900 x^2 - 200 x + 600\right) - \left(700 x^2 + 400 x\right) \][/tex]
Now, distribute the negative sign in the cost function:
[tex]\[ P(x) = -0.4 x^3 + 900 x^2 - 200 x + 600 - 700 x^2 - 400 x \][/tex]
Combine like terms:
[tex]\[ P(x) = -0.4 x^3 + (900 x^2 - 700 x^2) + (-200 x - 400 x) + 600 \][/tex]
[tex]\[ P(x) = -0.4 x^3 + 200 x^2 - 600 x + 600 \][/tex]
Therefore, the profit function is:
[tex]\[ P(x) = -0.4 x^3 + 200 x^2 - 600 x + 600 \][/tex]
This matches option A.
Thus, the correct answer is:
A. [tex]\( P(x) = -0.4 x^3 + 200 x^2 - 600 x + 600 \)[/tex]
