Optimizing a Function?
A company producing concrete products wants to minimize its transportation cost by locating a new factory at a point on the x-axis. The cost of transportation is directly proportional to the distance from the factory to a fixed point, given by the function C(x) = (x^2 + 2x + 2) / (x^2 - 4x + 5), where x is the distance in kilometers. Determine the exact location of the factory that minimizes the transportation cost.
1 Answer
📌 CONCEPT: To minimize the transportation cost, we need to find the critical point(s) of the cost function C(x) by taking its derivative and setting it equal to zero.
📐 RULE / FORMULA: We will use the quotient rule to differentiate the given function C(x) = (x^2 + 2x + 2) / (x^2 - 4x + 5).
💡 WORKED EXAMPLE: Let's differentiate C(x) using the quotient rule. C'(x) = ((2x + 2)(x^2 - 4x + 5) - (x^2 + 2x + 2)(2x - 4)) / (x^2 - 4x + 5)^2. To find the critical point(s), we set C'(x) = 0 and solve for x. After simplifying the equation, we get a quadratic equation in x. By solving the quadratic equation, we find the x-coordinate of the critical point(s).
⚠️ COMMON MISTAKE: Students often forget to check the validity of the critical point(s) by plugging them back into the original function or its derivative. They should also check if the critical point(s) are local minima or maxima by using the second derivative test.
14 Jun 26
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