91Ó°ÊÓ

In multivariable calculus, function analysis involves understanding how a function behaves, especially in relation to its variables and defined region. Here, we analyze the function \( f(x, y) = \frac{80}{xy} + 20y + 10x + 10xy \) within the region \( R \) where \( x, y > 0 \). This study is essential for investigating how the function reacts as its input values \( x \) and \( y \) change. By substituting different values into the function, we determine how each component contributes to the overall behavior of \( f(x, y) \). This includes:

• Understanding the influence of the terms like \( \frac{80}{xy} \), \( 20y \), \( 10x \), and \( 10xy \) as \( x \) and \( y \) vary.
• Recognizing which terms dominate as \( x \) or \( y \) become very large or very small, significantly affecting function values.

Function analysis is the foundation that helps us predict where the optimal values, such as minimums and maximums, may occur in the domain.

Critical points are the values of \( x \) and \( y \) where the partial derivatives of \( f(x, y) \) are zero, indicating potential minimums, maximums, or saddle points. Finding these involves calculating \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \), setting them to zero, and solving the resulting equations.For our function:- The partial derivative with respect to \( x \), \( \frac{\partial f}{\partial x} \), includes terms like \( -\frac{80}{x^2y} + 10y + 10 \).- The partial derivative with respect to \( y \), \( \frac{\partial f}{\partial y} \), includes terms like \( -\frac{80}{xy^2} + 20 + 10x \).Solving these equations helps locate the critical points. Critical points are pivotal as they guide us to pinpoint areas in the function where we might find extreme values. However, not all critical points will provide a global minimum or maximum, hence further analysis is necessary.

A global minimum is the lowest value of \( f(x, y) \) attainable for all \( x, y \) within a defined region. In this case, the function's behavior was tested at critical and boundary points. After analyzing how the function reacts when \( x \) and \( y \) are set to different extremes, we found:

• As \( x \) or \( y \) become extremely large or small, \( f(x, y) \) becomes much larger than 100.
• At point \((2, 1)\), \( f(x, y) = 100 \), which was verified to be smaller than the function's value at other tested points.

This comprehensive evaluation confirms that point \((2, 1)\) is the global minimum in region \( R \). Having a continuous function without pathways to negative infinity ensures such a global minimum must exist. Recognizing the global minimum is vital in optimization, as it represents the most efficient or least costly solution depending on the context.

The concepts of Continuity and Unboundedness are crucial in determining the existence of extreme values in multivariable calculus. A continuous function does not have abrupt changes, ensuring smoothness in the region:

• The function \( f(x, y) \) is continuous within \( R \), implying that as \( x \) and \( y \) change slightly, \( f(x, y) \) also changes smoothly.
• Unboundedness refers to the function's potential to reach infinitely large values if \( x \) or \( y \) approach boundary conditions like zero.

In this exercise, since \( f(x, y) \) does not reduce to negative infinity within the domain, it can have a global minimum. The conditions leading to such unbounded behavior explain why \( f(x, y) \) becomes significantly large outside specific boundaries. Continuity assures that there are no jumps or holes within \( R \), reinforcing the need to find a critical point where \( f(x, y) \) is minimized. Understanding these concepts helps guide the analysis to find not just any solution, but the best possible one.