the production function is given by f(x)= 4x^1/2.If the price of the commodity produced is $50 per unit and the cost of the input is $25 per unit, how much profit will the firm make if it maximize profits? A)$804 B)$385 C)$400 D)$198 E)$203

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Quizplus · Accepted Answer

The profit maximization problem for a firm involves choosing the level of input (x) to maximize the difference between total revenue and total cost. Given the production function $f(x) = 10x^{1/2}$, the price of the commodity ($50), and the cost of the input ($25), we can set up the profit function as follows:Profit = Total Revenue - Total Cost = $50*f(x) - $25*xSubstituting the production function into the profit equation gives:Profit = $50*10x^{1/2} - $25*x = $500x^{1/2} - $25*xTo maximize profit, we take the derivative of the profit function with respect to x, set it equal to zero, and solve for x:$\frac{d(Profit)}{dx} = \frac{d}{dx} (500x^{1/2} - 25x) = 250x^{-1/2} - 25 = 0$Solving for x gives:$250x^{-1/2} = 25$$x^{-1/2} = \frac{1}{10}$$x^{1/2} = 10$$x = 100$Substituting x = 100 back into the profit equation gives the maximum profit:Profit = $500*100^{1/2} - $25*100 = $500*10 - $2500 = $5000 - $2500 = $2500However, none of the provided options match this calculation. It appears there may have been a misunderstanding or misinterpretation of the question or the calculation process. Given the options, the correct answer based on the standard approach to such a problem cannot be determined as described. Please review the question and options for any discrepancies or additional context.

The Production Function Is Given by F(x)= 4x1/2

The Production Function Is Given by F(x)= 4x^1/2