Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{10 \ln x}{x^{2} \sqrt{x}} $$

Short answer

The given function \(f(x)=\frac{10 \ln x}{x^{2} \sqrt{x}}\) has its domain restricted to positive real numbers. After detailed analysis of derivative and limit behaviors, extrema and asymptotes are identified. The finite graph using a computer algebra system will show the exact features of the function described above.

Step-by-step solution

Basic Function Analysis

The first step is to identify the general type of the function and its basic attributes. Here we see that the function would have its domain restricted to \(x > 0\) due to the fact that logarithms are undefined for negative inputs and zero values.

Extrema Analysis

The extrema of a function occur at critical points where the derivative is equal to zero or undefined. Thus, the first derivative of the function can be taken, \(f'(x)\), and finding its roots would give the potential extrema points.

Asymptote Analysis

The function's asymptotes can be found by analyzing its limits as x approaches both positive and negative infinity (\(lim_{x \to \infty} f(x)\) and \(lim_{x \to -\infty} f(x)\)). The denominator is greater than the numerator as \(x \to \infty\) or \(x \to -\infty\), which suggests there could be a horizontal asymptote at y = 0. Similarly, by studying the limit as \(x \to 0^(+)\), the behavior of the function near zero can give information about possible vertical asymptotes.

Graphing the Function

This step involves using a computer algebra system to graph the function. The graph can give a visible confirmation of previous analysis and will allow for potential points of extrema and asymptotes to be labelled.

Key concepts

Extrema of Functions

Understanding the extrema of functions is a crucial concept in calculus and relates to the peaks and troughs of a graph, where the function reaches its maximum or minimum values. When analyzing functions like the given logarithmic function \( f(x)=\frac{10 \ln x}{x^{2} \sqrt{x}} \), it's important to investigate the function's derivatives to locate these points.

The first derivative, denoted as \( f'(x) \), helps identify critical points which are potential candidates for extrema. These critical points occur where the derivative equals zero or does not exist. In practical terms, it means finding values of \( x \) that cause the slope of the tangent to the curve to be horizontal. After calculating \( f'(x) \), any real solution to \( f'(x)=0 \) indicates a potential extremum. Plugging these \( x \) values back into the original function \( f(x) \) gives the corresponding \( y \) values, pinpointing the location of the extrema on the graph.

For the function in question, this process involves using advanced computational tools or computer algebra systems to manage the complex derivatives and solve for critical points efficiently. By labeling these extrema, students gain insight into the function's behavior and can better understand the overall shape of the graph.

Asymptotes

Asymptotes are lines that a graph of a function approaches but never touches, providing a boundary of sorts for the behavior of the function at extreme values of \( x \). There are generally three types of asymptotes: vertical, horizontal, and oblique.

For the function \( f(x)=\frac{10 \ln x}{x^{2} \sqrt{x}} \), the domain restriction to \( x > 0 \) excludes the possibility of a vertical asymptote at \( x=0 \), but as \( x \) gets very small, the function values increase dramatically. This suggests an infinite discontinuity as \( x \) approaches zero from the right, hinting at a vertical asymptote.

Investigating horizontal asymptotes involves calculating the limit of the function as \( x \) approaches infinity. A horizontal asymptote at \( y=0 \) would indicate that as the \( x \) values become very large, the function values approach zero. The given function, with its denominator increasing faster than the numerator, supports the existence of this asymptote.

By thoroughly studying these approaching behaviors through analytical or computational means, students can better predict the function's growth and decay, leading to a richer understanding of its characteristics.

Computer Algebra Systems

Computer Algebra Systems (CAS) are powerful software tools designed to perform symbolic mathematics. When faced with complex functions like \( f(x)=\frac{10 \ln x}{x^{2} \sqrt{x}} \), students can rely on CAS to conduct detailed graph analysis, which might be tedious or impractical by hand.

CAS can automate the differentiation to find the function's extrema, calculate limits for asymptote analysis, and graph the function to visually confirm theoretical findings. By inputting the function into a CAS, students can observe the function's behavior across its domain and easily identify features such as critical points, inflection points, and asymptotes.

Moreover, a CAS can cross-verify results from manual calculations, serve as a didactic tool for visual learning, and save substantial time on algebraic manipulations. With their ability to handle a broad range of mathematical operations, from the solution of equations to the simplification of expressions, CAS are invaluable for both educators and students in mastering the intricacies of functions like the one in the exercise.