Optimization of a Function?

📌 CONCEPT: The optimization of a function involves finding the maximum or minimum value of a function within a given constraint or condition, such as the volume of the hat in this case.

📐 RULE / FORMULA: To minimize the cost of production, we need to minimize the square of the radius (r^2) of the hat, given that the volume (V) of the hat is 200π cubic centimeters. The formula for the volume of a cone is V = (1/3)πr^2h, where h is the height of the cone.

💡 WORKED EXAMPLE: Given the volume V = 200π cubic centimeters, we can express the height h in terms of r as h = (3/π)*V/r^2 = (3/π)*(200π)/r^2 = 600/r^2. Now, the cost (C) is directly proportional to r^2, so C = kr^2 for some constant k. To minimize the cost, we need to find the critical point(s) by taking the derivative of C with respect to r and setting it equal to zero.

⚠️ COMMON MISTAKE: Students often forget to check the second derivative to confirm that the critical point corresponds to a minimum value.