Carmen's Detective Agency can use apprentice detectives,A,or experienced detectives,E.The cost of completing a job is C = 100 + A² + E² - 2AE.If a job requires a total of six detectives of either or both types,the cost-minimizing combination of detectives is: A) A = 3 and E = 3. B) A = 2 and E = 4. C) A = 4 and E = 2. D) A = 6 and E = 0. E) A = 0 and E = 6.

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The cost function is C = 100 + A^2 + E^2 - 2AE. To minimize the cost, we need to find the values of A and E that minimize this function.We can start by taking the partial derivative of C with respect to A:dC/dA = 2A - 2ESetting this equal to zero, we get:2A - 2E = 0A = EThis means that the number of apprentice detectives should be equal to the number of experienced detectives.We can now substitute this into the cost function to get:C = 100 + 2A^2 - 4A^2C = 100 - 2A^2To minimize this function, we need to find the value of A that minimizes A^2. This occurs when A = 0. However, since we need a total of six detectives, we cannot have A = 0. Therefore, the next best option is to have A = 1. This gives us E = 1, and the cost is C = 100 - 2(1)^2 = 98.We can now check the other answer choices to see if they give a lower cost.B) A = 2 and E = 4: C = 100 + 2(2)^2 + 4(4)^2 - 2(2)(4) = 100 + 8 + 64 - 16 = 156C) A = 4 and E = 2: C = 100 + 2(4)^2 + 2(2)^2 - 2(4)(2) = 100 + 32 + 8 - 16 = 124D) A = 6 and E = 0: C = 100 + 2(6)^2 + 0(0)^2 - 2(6)(0) = 100 + 72 - 0 = 172E) A = 0 and E = 6: C = 100 + 2(0)^2 + 6(6)^2 - 2(0)(6) = 100 + 0 + 216 - 0 = 316Therefore, the cost-minimizing combination of detectives is A = 1 and E = 1.

Carmen's Detective Agency Can Use Apprentice Detectives,A,or Experienced Detectives,E