# If the cost function and demand curve for a certain product are `C (x)= 60x + 7200` and `P(x)= 300 -2x` respectively: Find a. the total revenue function b. the marginal revenue function c. the marginal cost function d. point(s) at which there is breakeven. e.  the value of x at which marginal revenue equal to marginal cost. Show all work where required.

(a.) Total revenue function
Apply the formula,

Revenue = Price per unit `xx` Number of units sold

Take note that price per unit is the same as demand function. So,

`R(x) = (300-2x)*x`

`R(x)=300x-2x^2`

Hence, the revenue function is `R(x) = 300x-2x^2` .

(b.) Marginal revenue function
The formula for marginal revenue function is:

Marginal Revenue= R'(x)

So, take the derivative of R(x).

`R'(x) =(300x-2x^2)'`

`R'(x)=300-4x`

Thus, the marginal revenue function is `R'(x) =300-4x` .

(c.) Marginal cost function
To determine the marginal cost function, take the derivative of C(x).

Marginal Cost = C'(x)

`C'(x)=(60x + 7200)'`

`C'(x)=60`

Therefore, the marginal cost function is `C'(x) =60` .

d. Point(s) at which there is break-even.

To determine the break even point, set the cost and revenue functions equal to each other.

`C(x)=R(x)`

`60x+7200=300x-2x^2`

Then, set one side equal to zero.

`2x^2+60x-300x+7200=0`

`2x^2-240x +7200=0`

Simplify the equation and factor.

`x^2-120x+3600=0`

`(x-60)(x-60)=0`

And, set the factor equal to zero to get the value of x.

`x - 60 = 0`

`x=60`

Next, plug-in the value of x to either cost or revenue function.

`C(x) =60x+7200`

`C(60)=60(60)+7200`

`C(60)=10800`

Hence, the break-even point is (60,10800).

e. The value of x at which marginal revenue equal to marginal cost.

Set R'(x) and C'(x) equal.

`R'(x)=C'(x)`

`300-4x =60`

And, solve for x.

`-4x=60-300`

`-4x=-240`

`x=60`

Thus, the marginal revenue and marginal cost are equal when x=60.

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