find the maximum profit and number of untis that must be produced and sold in order to yield the maximum profit

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, Upper R left parenthesis x right parenthesisR(x), and cost, Upper C left parenthesis x right parenthesisC(x), are in thousands of dollars, and x is in thousands of unitsR(x)=6x-2x^2 , C(x)=x^3-3x^2+4x+1The production level for the maximum profit is about ____ units.

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Answer:

The production level for the maximum profit is about 1215 units.

Explanation:

The given income is R(x)= 6x-2x2 and the given cost is C(x)= x3-3x2+4x+1 where x is the quantity of units created and sold.

At that point the benefit work is P(x) = R(x)- C(x) = (6x-2x2) – (x3-3x2+4x+1) = - x3+x2+2x - 1.

The benefit will be most extreme when dP/dx is 0 and d2P/dx2 is negative. Here, dP/dx = - 3x2+2x+2 and d2P/dx2 = - 6x+2 .

On utilizing the quadratic recipe, on the off chance that dP/dx = 0, at that point x = [ - 2 ± √{ 22-4*(- 3)*2]/2*(- 3) = [-2 ± √(4+24)]/(- 6) = (2± √ 28)/6 = (1 ± √7)/3 . Since x can't be negative, consequently x = (1 + √7)/3 = 1.215250437 , state 1.215 ( on adjusting to the closest thousandth).

Likewise, d2P/dx2 is negative when x = 1.215, at that point .

Subsequently, the benefit will be greatest when 1215 units are created and sold.

The creation level for the greatest benefit is around 1215 units.