Chapter 8: Problem 43
Graph \(y=x^{1 / x}\) for \(x>0 .\) Show what happens for very small \(x\) and very large \(x\). Indicate the maximum value.
Short Answer
Expert verified
The function peaks at \( x = e \) with a maximum value of \( e^{1/e} \).
Step by step solution
01
Understand the function
The function is given by \( y = x^{1/x} \). This means for every value of \( x \), the exponent is the reciprocal of \( x \). We need to explore its behavior for different values of \( x \).
02
Consider Small Values of x
When \( x \) is very small, close to 0 (but positive), \( x^{1/x} \) approaches 0. This is because \( x^{1/x} = e^{\ln(x)/x} \) and \( \ln(x) \) is negative and very large for small \( x \), making \( \ln(x)/x \) very negative.
03
Consider Large Values of x
For large \( x \), \( x^{1/x} \) tends to 1 because \( \ln(x)/x \) approaches 0 as \( x \to \infty \). Hence, \( e^{\ln(x)/x} \to e^0 = 1 \).
04
Determine the Maximum Value
To find the maximum value, compute the derivative \( y'(x) \). Use the substitution \( y = e^{(\ln(x)/x)} \), then find the derivative using the chain rule and set it to zero to find critical points. Solve \( y'(x) = 0 \) to find that the maximum occurs around \( x = e \) (Euler's number), and the maximum value is \( e^{1/e} \).
05
Graph the Function
Using the information from the steps above, plot the function on a graph. The graph should show the function rising from 0, reaching a peak at \( x = e \), and gradually tapering off towards 1 as \( x \) increases.
06
Analyze and Interpret the Graph
On graphing the function, notice that \( y \) approaches 0 as \( x \) gets very small, reaches a maximum at \( x = e \), and approaches 1 as \( x \) increases further. This confirms the behavior of \( x^{1/x} \) as described in the previous steps.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus plays a significant role in understanding and analyzing the behavior of functions like the one given in this exercise. In calculus, we use derivatives to determine the rates of change and critical points of functions. This is crucial because it helps us find where a function's maximum or minimum values occur.
If we take the function in this exercise, \( y = x^{1/x} \), the use of derivatives allows us to explore how the function behaves as \( x \) changes. By calculating the derivative of \( y \) concerning \( x \), and setting it to zero, we can find the critical points. These points help us determine where the function reaches its maximum or minimum.
Beyond finding critical points, calculus helps us understand how the function approaches certain values as \( x \) becomes very small or very large. This is done using concepts like limits, which allow us to see that the function approaches 0 for small \( x \) and 1 for large \( x \). Calculus not only provides us the tools to find these points but also to check the concavity of the graph, providing a comprehensive understanding of its shape.
If we take the function in this exercise, \( y = x^{1/x} \), the use of derivatives allows us to explore how the function behaves as \( x \) changes. By calculating the derivative of \( y \) concerning \( x \), and setting it to zero, we can find the critical points. These points help us determine where the function reaches its maximum or minimum.
Beyond finding critical points, calculus helps us understand how the function approaches certain values as \( x \) becomes very small or very large. This is done using concepts like limits, which allow us to see that the function approaches 0 for small \( x \) and 1 for large \( x \). Calculus not only provides us the tools to find these points but also to check the concavity of the graph, providing a comprehensive understanding of its shape.
Exponential Functions
Exponential functions are critical to understanding the behavior of \( y = x^{1/x} \). In exponential functions, a variable appears in the exponent, which can lead to rapid increases or decreases in the function's value.
In our function, \( y = x^{1/x} \) can be rewritten as \( y = e^{\ln(x)/x} \). This form shows the relationship of the function as a transformation involving exponential growth rules. The use of Euler's number \( e \) here is pivotal because it simplifies finding derivatives and analyzing changes.
To comprehend the function's growth, we observe the component \( \ln(x)/x \). This term impacts how quickly or slowly the base of the exponential function changes as \( x \) changes. When \( x \) is extremely small, the logarithmic part \( \ln(x) \) becomes very negative, causing the exponential term \( e^{\ln(x)/x} \) to approach zero rapidly. Conversely, when \( x \) becomes large, \( \ln(x)/x \) tends towards zero, and since \( e^0 = 1 \), the function approaches 1. This analysis of exponential functions, thus, gives insights into both the management and consequences of the function's growth behavior.
In our function, \( y = x^{1/x} \) can be rewritten as \( y = e^{\ln(x)/x} \). This form shows the relationship of the function as a transformation involving exponential growth rules. The use of Euler's number \( e \) here is pivotal because it simplifies finding derivatives and analyzing changes.
To comprehend the function's growth, we observe the component \( \ln(x)/x \). This term impacts how quickly or slowly the base of the exponential function changes as \( x \) changes. When \( x \) is extremely small, the logarithmic part \( \ln(x) \) becomes very negative, causing the exponential term \( e^{\ln(x)/x} \) to approach zero rapidly. Conversely, when \( x \) becomes large, \( \ln(x)/x \) tends towards zero, and since \( e^0 = 1 \), the function approaches 1. This analysis of exponential functions, thus, gives insights into both the management and consequences of the function's growth behavior.
Critical Points
Critical points are fundamental in identifying where a function reaches its local maxima or minima, and they substantially influence the shape of the graph.
For \( y = x^{1/x} \), determining the critical points involves computing its derivative. The critical points occur where this derivative equals zero, signaling potential maxima or minima.
Using derivatives, the function \( y = e^{\ln(x)/x} \) is differentiated to find \( y'(x) \). Setting \( y'(x) = 0 \) helps us solve for \( x \), which in this exercise gives us \( x = e \) as the point of interest. At \( x = e \), we find the function's maximum value \( e^{1/e} \).
Understanding critical points further allows you to evaluate the second derivative to confirm if the point found is indeed a maximum by checking the function's concavity. Critical points don't just tell us about the extremal values but also assist in understanding how the function transitions and bends its path across the graph.
For \( y = x^{1/x} \), determining the critical points involves computing its derivative. The critical points occur where this derivative equals zero, signaling potential maxima or minima.
Using derivatives, the function \( y = e^{\ln(x)/x} \) is differentiated to find \( y'(x) \). Setting \( y'(x) = 0 \) helps us solve for \( x \), which in this exercise gives us \( x = e \) as the point of interest. At \( x = e \), we find the function's maximum value \( e^{1/e} \).
Understanding critical points further allows you to evaluate the second derivative to confirm if the point found is indeed a maximum by checking the function's concavity. Critical points don't just tell us about the extremal values but also assist in understanding how the function transitions and bends its path across the graph.
Graph Analysis
Graph analysis is a crucial step to visualize and make sense of the behavior of functions over different intervals.
The function \( y = x^{1/x} \) has distinct characteristics observable on a graph. By graphing, you can clearly see the function's rise, peak, and eventual leveling out.
For small values of \( x \), the graph shows \( y \) approaching zero quickly, as evident from the sharp upward climb. At \( x = e \), the function hits its peak, the maximum point. Beyond \( x = e \), the graph gradually slopes downward towards 1 as \( x \) increases, highlighting the exponential decay. This graphical behavior confirms our earlier calculus findings.
Graph analysis provides a visual representation of theoretical calculations and facilitates identification of trends such as asymptotic behaviors, slopes, intersections, and more. Creating a well-labeled graph enables clearer interpretation and communication of how functions operate over different domains.
The function \( y = x^{1/x} \) has distinct characteristics observable on a graph. By graphing, you can clearly see the function's rise, peak, and eventual leveling out.
For small values of \( x \), the graph shows \( y \) approaching zero quickly, as evident from the sharp upward climb. At \( x = e \), the function hits its peak, the maximum point. Beyond \( x = e \), the graph gradually slopes downward towards 1 as \( x \) increases, highlighting the exponential decay. This graphical behavior confirms our earlier calculus findings.
Graph analysis provides a visual representation of theoretical calculations and facilitates identification of trends such as asymptotic behaviors, slopes, intersections, and more. Creating a well-labeled graph enables clearer interpretation and communication of how functions operate over different domains.
