Analysis of the Unboundedness of a Function

In the context of mathematical functions, particularly in the realm of real analysis, it's essential to understand whether a function is bounded or unbounded. This analysis becomes crucial for various applications in science and engineering, where the behavior of functions at extreme values of their variables plays a significant role.

Understanding the Function

Consider the function ( f(x, y) e^{-x^2} e^{-4y} ). This function is a product of two exponential terms. To understand the behavior of this function, let's break it down first.

It can be re-expressed as:

( f(x, y) e^{-x^2} e^{-4y} )

This can be further simplified into two separate functions of one variable:

( f_1(x) e^{-x^2} )

( f_2(y) e^{-4y} )

Boundedness and Unboundedness

Let's analyze the behavior of these functions to determine if the original function is bounded or unbounded.

Analyzing ( f_1(x) e^{-x^2} )

The function ( f_1(x) e^{-x^2} ) is a well-known Gaussian function. As ( x ) varies over all real numbers, ( x^2 ) will always be non-negative, and as such, ( e^{-x^2} ) will always be positive. However, this function is bounded by 1, as:

( 0

Analyzing ( f_2(y) e^{-4y} )

The function ( f_2(y) e^{-4y} ) is an exponential function. As ( y ) tends to negative infinity, ( -4y ) tends to positive infinity, making the value of ( e^{-4y} ) also tend to positive infinity. This indicates that ( f_2(y) ) is not bounded.

Unboundedness of the Original Function

Given the nature of ( f_1(x) ) and ( f_2(y) ), the original function ( f(x, y) e^{-x^2} e^{-4y} ) inherits the unbounded behavior from ( f_2(y) ). Here's why:

( f(x, y) f_1(x) f_2(y) )

For any fixed ( x ), as ( y to -infty ), the term ( e^{-4y} ) will grow without bound, making the entire function also grow without bound. Mathematically, this can be expressed as:

( lim_{y to -infty} f(x, y) infty )

Proof of Unboundedness

To formally prove that the function ( f(x, y) ) is unbounded, we can show that for any constant ( M > 0 ), there exists a sequence of points ( (x_n, y_n) ) in ( mathbb{R}^2 ) such that:

( lim_{n to infty} |f(x_n, y_n)| infty )

For instance, consider the sequence ( x_n 0 ) and ( y_n -n ). As ( n to infty ), ( y_n ) tends to negative infinity, and thus:

( lim_{n to infty} f(0, y_n) lim_{y to -infty} f(0, y) lim_{y to -infty} e^{-4y} infty )

Continuous Function Analysis

Let's briefly discuss the continuity of the function. For a function with just one parameter, we know that it is continuous if it is differentiable at all points. For the function ( f(x, y) e^{-x^2} e^{-4y} ), we can use a similar approach. If both partial derivatives ( f_x(x, y) ) and ( f_y(x, y) ) are defined at all points, then ( f(x, y) ) is continuous at all points.

Note that this implication is not bidirectional. A function can be continuous without being differentiable at all points. Therefore, one must exercise caution when using this method to determine continuity.

Conclusion

In conclusion, the function ( f(x, y) e^{-x^2} e^{-4y} ) is unbounded because of the exponential term ( e^{-4y} ), which grows without bound as ( y ) approaches negative infinity. This analysis is crucial for understanding the behavior of such functions in various mathematical contexts.