Calculate the profit maximizing price of a monopolist firm with demand function `Q(x)=1-x` and cost of production `c(x)=x^2/2` `` ` `

In order to determine the maximum profit we have to must formulate a profit function: 

Profit function = Revenue Function - Cost Function 

Since we know what the cost function is, we are yet to determine the revenue function. 

Revenue, R(x), is determined by how much money is made by selling 'x' items and this is multiplied to the demand function. The revenue function is as follows: 

`R(x) = x* Q(x)`

`R(x) = x *(1-x)`

...

In order to determine the maximum profit we have to must formulate a profit function: 

Profit function = Revenue Function - Cost Function 

Since we know what the cost function is, we are yet to determine the revenue function. 

Revenue, R(x), is determined by how much money is made by selling 'x' items and this is multiplied to the demand function. The revenue function is as follows: 

`R(x) = x* Q(x)`

`R(x) = x *(1-x)`

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

Since we have the Revenue function we can now mathematically formulate the Profit function P(x): 

`P(x) = R(x) - c(x)`

`P(x) = x -x^2 - (x^2/2)`

`P(x) = x - ((2x^2)/2) - (x^2/2)`

`P(x) = x - ((3x^2)/2)`

Now we have a profit function. We can now apply calculus to determine the maximum profit by finding the first derivative: 

`(dP)/dx = 1 -(6x)/2`

`(dP)/dx = 1-3x`

Then. we make the first derivative equal zero in order to determine the maximum:

`0 = 1 -3x`

`3x = 1`

`x = 1/3` ` `

Now we know the value of x when the profit is a maximum, now we substitute this back in the Profit function: 

`P(1/3) = 1/3 - ((3/2) *(1/3)^2)`

`P = 1/6`