Maxima and Minima on a Rational Function?

📌 CONCEPT: The problem asks us to determine if the absolute maximum and local minimum points of a rational function are also points of inflection.

📐 RULE / FORMULA: To find points of inflection, we need to solve the equation f''(x) = 0, where f''(x) is the second derivative of the function.

💡 WORKED EXAMPLE: Consider the function f(x) = (x^2 - 1) / (x^2 + 1). First, we need to find its first and second derivatives. Then, we set the second derivative equal to zero and solve for x to find the points of inflection. After that, we can compare these points with the points where the function has an absolute maximum and a local minimum.

⚠️ COMMON MISTAKE: Students may incorrectly solve the equation f''(x) = 0, or fail to compare the points of inflection with the maximum and minimum points, leading to incorrect conclusions.