Two stores are located side by side and attract customers to each other and to themselves by advertising. Where x₁ and x₂ are the advertising expenditures of stores 1 and 2, the profits of the firms are (48 + x₂)x₁ - 2(x₁)² for store 1 and (54 + x₁)x₂ - 2(x₂)² for store 2. Knowing these functions, one investor buys both stores. In order to maximize his total profits, how much should he spend on advertising for store 1? A) $10 B) $26 C) $25 D) $35 E) None of the above.

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Quizplus · Accepted Answer

To maximize total profits, the investor should choose x₁ such that the derivative of the sum of the two profit functions with respect to x₁ is equal to zero. Taking the derivative of the sum of the profit functions with respect to x₁ and setting it equal to zero, we get: 48 + 2x₂ - 2² = 54 - 2x₂ - 2² Simplifying and solving for x₂, we get: x₂ = 3 Substituting this value of x₂ into either profit function gives us the profit of the two stores combined: Profit = (48 + 3)(x₁) - 2(x₁)² + (54 + 3)(1 - x₁) - 2(1 - x₁)² Simplifying this, we get: Profit = 102 - 6x₁ - 2² To maximize profit, we need to choose x₁ such that the derivative of this expression with respect to x₁ is equal to zero. Taking the derivative and setting it equal to zero, we get: -6 = 0 This is obviously not true, so there is no value of x₁ that maximizes the total profit. However, we can still find the value that maximizes the profit of store 1 alone, given that we're choosing x₂ = 3. To maximize the profit of store 1, we need to choose x₁ such that the derivative of the profit function of store 1 with respect to x₁ is equal to zero. Taking the derivative and setting it equal to zero, we get: 48 + 3x₂ - 2² = 0 Solving for x₁, we get: x₁ = (2² - 48) / 3 Substituting x₂ = 3 and ² = 4, we get: x₁ = (2*4 - 48) / 3 = -12/3 = -4 Since advertising expenditure cannot be negative, we choose the value of x₁ that is closest to -4 but still non-negative, which is x₁ = 1. Therefore, the investor should spend $25 on advertising for store 1.

Two Stores Are Located Side by Side and Attract Customers