Question

Examine the function for extrema without using the derivative tests, and use a computer algebra system to graph the surface. (Hint: By observation, determine if it is possible for \(z\) to be negative. When is \(z\) equal to \(0 ?)\) $$ z=\frac{\left(x^{2}-y^{2}\right)^{2}}{x^{2}+y^{2}} $$

Short answer

The function \(z\) equals zero along the lines \(x=y\) and \(x=-y\). It is not possible for \(z\) to be negative. The graph of the function using a computer algebra system confirms this.

Step-by-step solution

Understand Behavior with respect to Zero

Try to understand when \(z\) equals zero. For this, we need \(x^2 - y^2 = 0\). This yields a set of solutions where \(x = y\) or \(x = - y\). Thus, \(z\) is equal to zero along the lines \(x=y\) and \(x= -y\).

Determine Whether \(z\) can be Negative

Next, determine if it is possible for \(z\) to be negative. Given that \(z\) is a proportion of two squared values, it is not possible for \(z\) to be negative because a square of any real number is always positive or zero. Therefore, \(z\) cannot be negative.

Use Computer Algebra System to Graph

Finally, use a computer algebra system to graph the given function, \(z=\frac{\left(x^{2}-y^{2}\right)^{2}}{x^{2}+y^{2}}\). This will help in visually observing the function's behavior, showing its graph above the lines \(x = y\) and \(x = - y\), and that it does not take any negative values.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is a powerful software tool designed to perform complicated mathematical operations. It can solve equations, simplify expressions, perform integrations, find limits and derivatives – and that's just scratching the surface. In our context, a CAS is a valuable asset for graphing functions, particularly when dealing with complex multivariable functions like the one in our exercise.

Using a CAS to graph the function \( z=\frac{\left(x^{2}-y^{2}\right)^{2}}{x^{2}+y^{2}} \) makes it easier to visualize extrema, intercepts, and contours of the function. With a CAS, you can rotate and zoom in/out of the graph, offering a detailed view from various perspectives. Plus, in cases where the derivative tests are not utilized, as suggested in the exercise, a CAS provides an alternative visual method to analyze functions. By seeing the behavior of the function graphically, students can improve their understanding and gain intuition on the shape and properties of the function.

Graphing Multivariable Functions

Graphing multivariable functions, like our function of \( z \), is a more complex process compared to single-variable functions. The reason is that multivariable functions have surfaces in a three-dimensional space. To graph such functions, you need to understand the roles of \( x \) and \( y \) coordinates in shaping the surface.

Fortunately, graphing software and CAS simplify this process. These systems can take an equation involving two variables and create a 3D visualization. For our function, the CAS would show a surface above the \( xy \) -plane, with contours changing as \( x \) and \( y \) change. Specifically, for the function \( z=\frac{\left(x^{2}-y^{2}\right)^{2}}{x^{2}+y^{2}} \), by observing the graph, students can see that \( z \) equals zero along the lines where \( x = y \) or \( x = - y \) without having to perform complex calculations.

Analyzing Function Behavior

Analyzing the behavior of functions, particularly without using derivative tests, relies heavily on understanding the properties of equations and being able to deduce information from them. In the given exercise, one key insight is recognizing that \( z \) cannot be negative because it is the quotient of squares. Squares always yield non-negative results as the square of any real number is positive or zero.

Furthermore, by observing that \( z \) is equal to zero when \( x^{2} - y^{2} = 0 \) gives us critical information about the nature of the graph at certain points. In the context of our function, this tells us that the surface crosses the \( xy \) -plane along lines where \( x \) and \( y \) are equal and opposite. By combining these analytical skills with graphing in a CAS, students can form a comprehensive understanding of a function's behavior in a more intuitive and less mechanically intense way, which is particularly useful when tackling complex calculus or higher-dimensional problems.