Optimization of a Function?

📌 CONCEPT: The problem of minimizing the total cost of production involves optimizing a function by finding the minimum value of the cost function.

📐 RULE / FORMULA: To optimize a function, we need to find the critical points by taking the derivative of the function and setting it equal to zero, and then determine the nature of these points using the second derivative test.

💡 WORKED EXAMPLE: Given the cost function C(x) = 2x^2 + 16x + 25, we first find its derivative: C'(x) = 4x + 16. Setting C'(x) = 0, we get 4x + 16 = 0, which gives x = -4. However, since x represents the number of boxes produced, it cannot be negative. Therefore, we ignore this critical point and find the second derivative C''(x) = 4, which is positive, indicating that the function has a minimum value at x = 0. Adding the fixed cost of $100, the total cost is minimized when no boxes are produced.

⚠️ COMMON MISTAKE: Students often ignore the domain of the function and consider negative values of x, which may not be physically meaningful in the context of the problem.