Find the critical point(s) for the function f(x, y) = 2x^2 + y^2 - 4x + 6y - 5.

Question

Find the critical point(s) for the function f(x, y) = 2x^2 + y^2 - 4x + 6y - 5. Rose's Confections manufactures and sells x pounds of fudge per week. The monthly cost and price-demand equations are C(x) = 1,500 + 4x p(x) = 40 - x/3 where C is the total cost of producing x pounds of fudge that sell for P dollars each. a) Find the revenue function. b) Find the marginal revenue function. c) Use the marginal revenue function to approximate the additional revenue that would result from selling the 52nd pound of fudge. d) Find the (exact) additional revenue that would result from selling the 52nd pound of fudge.

Explanation / Answer

Revenue = price * quantity

R(x) = x * p(x) = 40x - [x^2]/3

b.) Find the marginal revenue function

MR = R'(x) = 40 - [2x]/3

c.) Use the marginal revenue function to approximate the additional revenue (for the 52nd pound of fudge)

I took the average of the surrounding points (53rd pound and 51st pound)

[R'(53) - R'(51) ] / 2 = $2,084.67 - $2,006 / 2 = $2,045.34


d.) Find the exact additional revenue (for the 52nd pound)

MR = R'(52) = $2,045.33

[MR(53) - MR(51)] / 2 follows the estimation of the first derivative which you can find on this webpage

df / dx = f(x+h) - f(x-h) / 2h

(in part c.) I used the formula and set h = 1)

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