Question

Absolute extrema on open and/or unbounded regions If possible, find the absolute maximum and minimum values of the following functions on the region \(R\). $$f(x, y)=x^{2}-y^{2} ; R=\\{(x, y):|x| < 1,|y| < 1\\}$$

Short answer

Are there any absolute maximum or minimum values? Answer: The critical point $(0, 0)$ is a saddle point. There are no absolute maximum or minimum values for the function $f(x, y) = x^2 - y^2$ on the given region, as the region is open, and the boundary values are not achievable.

Step-by-step solution

Calculate the partial derivatives

Calculate the partial derivatives of the function \(f(x,y) = x^2 - y^2\) with respect to \(x\) and \(y\): $$ \frac{\partial f}{\partial x} = 2x \\ \frac{\partial f}{\partial y} = -2y $$

Solve for critical points

To find the critical points, set both partial derivatives equal to zero and solve for \(x\) and \(y\): $$ 2x = 0 \Rightarrow x = 0 \\ -2y = 0 \Rightarrow y = 0 $$ So the critical point is at \((0,0)\).

Analyze the critical point

To determine the nature of the critical point, we should check if it's a maximum, minimum, or saddle point. An alternative to using the second derivative test is to observe that the function \(f(x, y) = x^2 - y^2\) is a hyperbolic paraboloid, which means its critical point at \((0, 0)\) represents a saddle point. Thus, our critical point does not give us an absolute maximum or minimum value.

Evaluate the function on the boundary

To find the extrema, we need to evaluate the function on the boundary of the region. Since the region is open, the boundary points cannot be included. However, we can still approach them close enough to get their limit values. On the boundary, the values of the function can be written as follows: $$ \lim_{x\to \pm1^-}f(x,y) = \pm y^2 - 1 \ \Rightarrow \ \text{when} \ |y| < 1 \\ \lim_{y\to \pm1^-}f(x,y) = x^2 - 1 \ \Rightarrow \ \text{when} \ |x| < 1 $$

Determine the absolute maximum and minimum

Now, we need to compare the values of the function on the boundary and at the saddle point: 1. At the saddle point \((0,0)\), \(f(0,0) = 0^2 - 0^2 = 0\). 2. On the boundary with \(x = \pm 1\) and \(|y| < 1\), the function value can range from \(-\infty\) to \(1\) (exclusive). For example, as \(y \to 0^-\), \(f(1,y) \to 1^-\). 3. On the boundary with \(y = \pm 1\) and \(|x| < 1\), the function value can range from \(-1\) to \(\infty\) (exclusive). For example, as \(x \to 0^-\), \(f(x,1) \to -1^+\). Hence, the absolute minimum value of the function on the region R is not achievable, while the absolute maximum value is also not achievable due to the open region. Even though the boundary's values approach specific values, they are never actually achieved.

Key concepts

Open Region

An open region in the context of functions and calculus refers to an area in the coordinate plane that does not include its boundary. When we talk about the region \( R = \{(x, y) : |x| < 1, |y| < 1 \} \), we are describing a square excluding its edges. This can be likened to a sandbox without borders, allowing free movement within but not on the lines defining its limits.

Open regions are significant when we look for extrema, such as maximum and minimum values. Unlike closed regions, where you might find extrema on the boundary, open regions require different approaches. Specifically, exploring potential extrema means working within the proximity of the boundaries without ever having access to those boundary points.

In practical terms, while approaching the edges can show us trends or limits, absolute extremum values may not be found because they 'stretch' towards the edge, rather than being fixed at a boundary.

Partial Derivatives

Partial derivatives represent the rate of change of a function concerning one variable, holding the others constant. For a function of two variables, such as \( f(x, y) = x^2 - y^2 \), partial derivatives help us understand how changes in one variable influence the whole function, ignoring the contribution of the other.

To calculate the partial derivative with respect to \( x \), differentiate \( f(x, y) \) treating \( y \) as a constant:

• \( \frac{\partial f}{\partial x} = 2x \)

Similarly, with respect to \( y \), treating \( x \) as a constant:

• \( \frac{\partial f}{\partial y} = -2y \)

The results tell us that, at any point, the function's slope in the \( x \)-direction is proportional to \( x \), whereas in the \( y \)-direction, it's proportional to the negative of \( y \).

These derivatives are fundamental for finding critical points and determining the behavior of functions in multivariable calculus.

Critical Points

Critical points of a function in calculus are where its partial derivatives are zero or undefined. These points are crucial as they help identify where a function might have a local maximum, local minimum, or a saddle point. For the function \( f(x, y) = x^2 - y^2 \), setting the partial derivatives equal to zero, we found the critical point is \( (0, 0) \).

• \( \frac{\partial f}{\partial x} = 0 \Rightarrow 2x = 0 \Rightarrow x = 0 \)
• \( \frac{\partial f}{\partial y} = 0 \Rightarrow -2y = 0 \Rightarrow y = 0 \)

At this point, you might typically apply a second derivative test to determine the nature of the critical point, but sometimes functions can have properties that provide immediate insight.

In this example, since \( f(x, y) = x^2 - y^2 \) forms a hyperbolic paraboloid, the critical point is a saddle point. Saddle points are neither maxima nor minima, showing where one part of the function increases while the other decreases, geometrically resembling a saddle or a trampoline.

Hyperbolic Paraboloid

A hyperbolic paraboloid is a type of three-dimensional surface, often resembling a saddle. It is defined by functions like \( f(x, y) = x^2 - y^2 \). The unique shape creates an interesting surface, curving upwards along one axis and downwards along the other. This gives it the distinctive 'saddle' appearance.

When a function has a form like \( x^2 - y^2 \), the resulting graph is a hyperbolic paraboloid. It means that at any critical point, like \( (0, 0) \), the function acts as a saddle point. Neither a maximum nor minimum can be clearly determined at such points because of the surface's shape.

• Along one path, the surface rises, contributing to a local peak.
• Conversely, along another path, it falls, contributing to a local trough.

When studying functions over an open region, recognizing hyperbolic paraboloid behavior can be critical. This understanding helps in explaining why the absolute extrema are not attained at saddle points within an open region. The nature of the surface affects how we interpret the critical points encountered in functions.