Exploring the Behavior of f(xy) 1 - 1/(1xy): Maximum, Minimum, and Unbounded Values
The function ( f(xy) 1 - frac{1}{1xy} ) exhibits interesting properties in terms of its behavior with respect to (xy). Understanding its maximum and minimum values, as well as whether it is bounded or unbounded, is crucial in many mathematical and practical applications. In this article, we will delve into the details of this function and analyze its key characteristics.
Behavior of the Function
Let's start by examining the function ( f(xy) 1 - frac{1}{1xy} ). We will analyze the function's behavior by investigating its critical points and the nature of these points.
Maximum and Minimum Values
First, consider the conditions under which ( f(xy) ) reaches its maximum or minimum values. The function can be rewritten as:
[f(xy) 1 - frac{1}{1xy}]We need to analyze the term (-frac{1}{1xy}), which is dependent on (xy). The expression (1xy) is always positive (assuming (x) and (y) are real and non-zero), and thus [-frac{1}{1xy}] will take values that decrease without bound as (xy) increases. Conversely, as (xy) approaches (-1), (-frac{1}{1xy}) approaches (infty).
Therefore, the function ( f(xy) ) has no finite maximum or minimum values. Specifically:
Maximum: ( f(xy) ) does not have a maximum value because as (xy to -1), (-frac{1}{1xy} to infty), leading to ( f(xy) to infty ). Minimum: ( f(xy) ) does not have a minimum value because (-frac{1}{1xy} to -infty) as (xy to -1), leading to ( f(xy) to -infty ).Analysis at Critical Points
To further determine the nature of critical points, we need to analyze the partial derivatives of ( f(xy) ).
[f(xy) 1 - frac{1}{1xy}]First, compute the partial derivatives:
[frac{partial f}{partial x}(xy) frac{y}{1xy^2}] [frac{partial f}{partial y}(xy) frac{x}{1xy^2}]Evaluating these partial derivatives, we find that the only critical point occurs when both partial derivatives are zero:
[frac{partial f}{partial x}(xy) 0 quad text{and} quad frac{partial f}{partial y}(xy) 0]This occurs at (x 0) and (y 0), which is the point ((0, 0)).
Hessian Matrix Analysis
To classify the critical point ((0, 0)), we need to compute the second-order partial derivatives and the Hessian matrix.
[frac{partial^2 f}{partial x^2}(xy) -frac{y^2}{1xy^3}] [frac{partial^2 f}{partial x partial y}(xy) frac{1-xy}{1xy^3}] [frac{partial^2 f}{partial y^2}(xy) -frac{x^2}{1xy^3}]At the critical point ((0, 0)), the Hessian matrix is:
[H begin{bmatrix} 0 1 1 0 end{bmatrix}]The determinant of the Hessian matrix is:
[text{det}(H) (0)(0) - (1)(1) -1]Since the determinant of the Hessian matrix is negative, the critical point ((0, 0)) is an saddle point, which means the function is not bounded around this point and oscillates between positive and negative values.
Conclusion
In conclusion, the function ( f(xy) 1 - frac{1}{1xy} ) is unbounded and does not have finite maximum or minimum values. The only critical point, ((0, 0)), is a saddle point, indicating that the function behaves in an oscillatory manner around this point.