Question

# The revenue function of a company is given by R(x)=-2x^2+25x+150, the cost function is given by...

The revenue function of a company is given by R(x)=-2x^2+25x+150, the cost function is given by C(x)=13x+100

a. Find the marginal cost and marginal revenue function.

b. Find the production level x where the profit is maximized. Then find the maximum profit.

Requirement (a):

Marginal Cost function = Derivative of x for C(x) = 13X + 100 i.e. 13 [MC(x) = 13]

Marginal Revenue Function = Derivative of x for R(x) = -2x^2 + 25x + 150 i.e. -4x + 25 [MR(x) = -4x + 25]

Requirement (b):

Profit is maximum at a level where marginal revenue is equal to marginal cost (MR = MC)

i.e. -4x + 25 = 13

-4x = 13 - 25

-4x = -12

x = 3

The production level at which profit is maximum is 3 units

Revenue : R(x=3) = -[2*(3^2)] + (25 * 3) + 150

R (3) = -18 + 75 + 150

R (3) = 207

Cost : C(x=3) = (13 * 3) + 100

C (3) = 139

Maximum Profit = R (x=3) - C (x=3)

= 207 - 139 i.e. 68

The maximum profit is \$68.

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