If
C(x) = 12000 + 400x − 1.6x2 + 0.004x3
is the cost function and
p(x) = 1600 − 7x
is the demand function, find the production level that will maximize profit.
profit = revenue - cost = quantity*price - cost
f(x) = x*p(x) - c(x)
= x(1600-7x) - (12000 + 400x − 1.6x^2 + 0.004x^3)
Now just find x>0 where f'(x) = 0
To find the production level that will maximize profit, we need to determine the profit function first. The profit function is given by:
Profit (P) = Revenue (R) - Cost (C)
The revenue function can be calculated by multiplying the demand function, p(x), by the production level, x:
R(x) = p(x) * x
Plugging in the given demand function, we have:
R(x) = (1600 - 7x) * x
= 1600x - 7x^2
Now, we can rewrite the profit function using the revenue and cost functions:
P(x) = R(x) - C(x)
= (1600x - 7x^2) - (12000 + 400x - 1.6x^2 + 0.004x^3)
= -0.004x^3 + 5.6x^2 + 1200x - 12000
To find the production level that maximizes profit, we need to find the value of x that maximizes the profit function P(x). This can be done by finding the critical points of the function, which are the values of x where the derivative of P(x) equals zero.
To get the derivative of P(x), we can differentiate the profit function with respect to x:
P'(x) = d/dx (-0.004x^3 + 5.6x^2 + 1200x - 12000)
= -0.012x^2 + 11.2x + 1200
Next, we set the derivative equal to zero and solve for x:
-0.012x^2 + 11.2x + 1200 = 0
To solve this quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -0.012, b = 11.2, and c = 1200. Plugging in these values, we can calculate the solutions for x.
Once you have the critical points, you can evaluate the profit function at these points and identify the production level that maximizes profit.
To find the production level that will maximize profit, we need to calculate the profit function. The profit function can be calculated by taking the difference between the revenue function and the cost function.
Revenue function, R(x) = p(x) * x
Profit function, P(x) = R(x) - C(x)
Now, let's substitute the given demand function and cost function into the profit function:
P(x) = (1600 - 7x) * x - (12000 + 400x - 1.6x^2 + 0.004x^3)
Next, we need to find the derivative of the profit function with respect to x and set it equal to zero to find the critical points. The critical points will help us determine the production level that maximizes profit.
dP/dx = 0
Let's find the derivative of the profit function:
dP/dx = 1600 - 7x - (12000 + 400x - 1.6x^2 + 0.004x^3)
Simplifying the equation:
dP/dx = 1600 - 7x - 12000 - 400x + 1.6x^2 - 0.004x^3
Now set the derivative equal to zero and solve for x:
1600 - 7x - 12000 - 400x + 1.6x^2 - 0.004x^3 = 0
Let's solve this equation to find the critical points.