This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the practice problems. 41. Find the value of x that maximizes y = 20x - x^2. Step 1: To find the maximum value, we take the first derivative of the function y with respect to x. (dy)/(dx) = (d)/(dx)(20x - x^2) = 20 - 2x Step 2: Set the first derivative equal to zero and solve for x. 20 - 2x = 0 2x = 20 x = 10 The value of x that maximizes y is 10. 42. Find the value of x that minimizes y = x^2 + 4x + 7. Step 1: To find the minimum value, we take the first derivative of the function y with respect to x. (dy)/(dx) = (d)/(dx)(x^2 + 4x + 7) = 2x + 4 Step 2: Set the first derivative equal to zero and solve for x. 2x + 4 = 0 2x = -4 x = -2 The value of x that minimizes y is -2. 43. Given TC = x^2 + 10x + 50, find the Marginal Cost. Step 1: Marginal Cost (MC) is the derivative of the Total Cost (TC) function with respect to x. MC = (d(TC))/(dx) Step 2: Differentiate the given TC function. MC = (d)/(dx)(x^2 + 10x + 50) = 2x + 10 The Marginal Cost is 2x + 10. 44. Given TR = 100x - x^2, find the Marginal Revenue. Step 1: Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to x. MR = (d(TR))/(dx) Step 2: Differentiate the given TR function. MR = (d)/(dx)(100x - x^2) = 100 - 2x The Marginal Revenue is 100 - 2x. 45. A firm's profit function is = 120x - x^2 - 200. Determine the output level that maximizes profit. Step 1: To maximize profit, we take the first derivative of the profit function with respect to x. (d)/(dx) = (d)/(dx)(120x - x^2 - 200) = 120 - 2x Step 2: Set the first derivative equal to zero and solve for x. 120 - 2x = 0 2x = 120 x = 60 The output level that maximizes profit is 60. Send me the next one 📸