To determine the profit function, [tex]\( P(x) \)[/tex], we need to subtract the cost function, [tex]\( C(x) \)[/tex], from the revenue function, [tex]\( R(x) \)[/tex]. Given the cost function and revenue function:
[tex]\[ C(x) = 500x^2 + 400x \][/tex]
[tex]\[ R(x) = -0.6x^3 + 800x^2 - 300x + 600 \][/tex]
The profit function [tex]\( P(x) \)[/tex] is defined as:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions into the equation:
[tex]\[ P(x) = (-0.6x^3 + 800x^2 - 300x + 600) - (500x^2 + 400x) \][/tex]
Now, distribute the negative sign and combine like terms:
[tex]\[ P(x) = -0.6x^3 + 800x^2 - 300x + 600 - 500x^2 - 400x \][/tex]
Combine the [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex] terms:
[tex]\[ P(x) = -0.6x^3 + (800x^2 - 500x^2) + (-300x - 400x) + 600 \][/tex]
Simplify further:
[tex]\[ P(x) = -0.6x^3 + 300x^2 - 700x + 600 \][/tex]
So, the correct profit function is:
[tex]\[ P(x) = -0.6x^3 + 300x^2 - 700x + 600 \][/tex]
Hence, the correct answer is:
B. [tex]\( P(x) = -0.6x^3 + 300x^2 - 700x + 600 \)[/tex]
