The cost of producing commodity is C(X)=3X^2+4X+8 dollars.If the price is P(X)=(50-X) dollars per unit,determine the level of production that maximizes the profit.

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What is the level of production.

profit=(price-cost)x

= 50x-x^2 -3x^3-4x^2-8x

dprofit/dx=0= 50-2x-9x^2-8x-8

9x^2-10x+42=0

x= (10+sqrt(10000-1512))/18 =5.6
since you cant produce .6 of the units, try x=6, and x=5 to see which is the greater profit.

check my thinking and work.

To determine the level of production that maximizes profit, we first need to understand the components of profit and how it is calculated.

Profit is the difference between total revenue and total cost. Total revenue is calculated by multiplying the price per unit (P(X)) by the quantity produced (X). So, total revenue (R(X)) is given by R(X) = P(X) * X.

Total cost (C(X)) is already given as C(X) = 3X^2 + 4X + 8.

Profit (P(X)) is calculated as P(X) = R(X) - C(X).

Now, we can express the profit function in terms of X:

P(X) = (50 - X) * X - (3X^2 + 4X + 8).

To find the level of production that maximizes profit, we need to find the value of X that maximizes the profit function P(X).

To do this, we can take the derivative of the profit function with respect to X and set it equal to zero to solve for X:

P'(X) = 0.

To find the derivative of P(X), we differentiate each term separately:

P'(X) = (50 - 2X) - (6X + 4) = 50 - 2X - 6X - 4 = -8X + 46.

Now, set P'(X) = 0 and solve for X:

-8X + 46 = 0,
-8X = -46,
X = -46 / -8,
X = 5.75.

So, the level of production that maximizes profit is approximately 5.75 units.