h) A monopolist faces the demand function given by x=10-0.5p where x is the number of units produced and sold at price p dollars per unit. The total cos incurred consists of a fixed overhead of 28 dollars plus production cost of 2 dollars per unit of x. i Write down the monopolist's profit as a function of x (2 Marks) ii Find the maximum profit realized by the monopolist in producing x units of the commodity (2 Marks)

i) To write down the monopolist's profit as a function of x, we need to consider the revenue and cost.<br /><br />Revenue is calculated by multiplying the price per unit (p) by the number of units sold (x). From the given demand function, we can express p as a function of x: p = 10 - 0.5x.<br /><br />Revenue = p * x = (10 - 0.5x) * x = 10x - 0.5x^2<br /><br />The total cost incurred by the monopolist consists of a fixed overhead of 28 dollars plus a production cost of 2 dollars per unit of x.<br /><br />Total cost = Fixed overhead + Production cost = 28 + 2x<br /><br />Now, we can write the profit function as the difference between revenue and total cost:<br /><br />Profit = Revenue - Total cost = (10x - 0.5x^2) - (28 + 2x) = 10x - 0.5x^2 - 28 - 2x = 8x - 0.5x^2 - 28<br /><br />So, the monopolist's profit as a function of x is: Profit = 8x - 0.5x^2 - 28<br /><br />ii) To find the maximum profit realized by the monopolist, we need to find the value of x that maximizes the profit function.<br /><br />To do this, we can take the derivative of the profit function with respect to x and set it equal to zero:<br /><br />d(P)/dx = 8 - x = 0<br /><br />Solving for x, we get x = 8.<br /><br />Now, we can substitute x = 8 into the profit function to find the maximum profit:<br /><br />Profit = 8(8) - 0.5(8)^2 - 28 = 64 - 32 - 28 = 4<br /><br />Therefore, the maximum profit realized by the monopolist in producing 8 units of the commodity is 4 dollars.