Exploring the Global Extrema of the Function y x / (x^2 - 1)

Understanding the behavior of functions is a crucial part of mathematical analysis. In this article, we will explore the global extrema of the function y x / (x^2 - 1). We will find the critical points and classify the global extrema of the function, providing a clear and detailed explanation of each step.

Introduction to the Function

The function in question is y x / (x^2 - 1). This function is interesting due to its structure and the potential for multiple maximum and minimum points within its domain. Our objective is to find the global maximum and minimum values of this function and determine their locations.

Graphical Analysis

Before diving into the analytical approach, it is useful to visualize the function using graphs. From the graphical analysis, we can estimate the locations of the maximum and minimum values. In this case, the function's behavior suggests that the maximum value is approximately at x ≈ 0.4142.

Analytical Solution

Step 1: Finding the Critical Points

First, we need to find the critical points of the function by setting its derivative equal to zero.

The derivative of the function y x / (x^2 - 1) is given by:

y' (x^2 - 1) - 2x^2 / (x^2 - 1)^2 1 / (x^2 - 1) - 2x / (x^2 - 1)^2

Step 2: Solving for Critical Points

Setting the derivative equal to zero and solving for x, we get:

1 / (x^2 - 1) - 2x / (x^2 - 1)^2 0

This simplifies to:

(x^2 - 1) - 2x 0

Which further simplifies to:

x^2 - 2x - 1 0

Solving this quadratic equation, we get:

x -1 - sqrt(2) or x sqrt(2) - 1

Step 3: Evaluating the Function at the Critical Points

Evaluating the function at the critical points:

At x -1 - sqrt(2): y -0.207107

At x sqrt(2) - 1: y 1.20711

Step 4: Determining the Global Extrema

Considering the endpoints of the domain, we see that at x -∞ and x ∞, the function approaches zero.

Thus, we can classify the extrema as follows:

x -1 - sqrt(2): global minimum

x sqrt(2) - 1: global maximum

Conclusion

In conclusion, the function y x / (x^2 - 1) has a global minimum at x -1 - sqrt(2) with a value of approximately -0.207107 and a global maximum at x sqrt(2) - 1 with a value of approximately 1.20711.

Understanding these critical points and extremum values is crucial for a deeper understanding of the function's behavior and can be applied in various fields, including physics, engineering, and economics. This analysis method, combining graphical and analytical approaches, provides a comprehensive solution to finding the extrema of complex functions.

Keywords: function optimization, critical points, global extrema