Question

In Exercises, graph and analyze the function. Include any relative extrema and points of inflection in your analysis. Use a graphing utility to verify your results. $$ y=\frac{x}{\ln x} $$

Short answer

The function \(y = \frac{x}{\ln x}\) has an undefined point at \(x = 1\). It has a relative maximum at \(x = e\). The function does not have any points of inflection.

Step-by-step solution

Find the Undefined Points

Firstly, find where the function is undefined. This function will be undefined when the denominator \(\ln x = 0\), which yields \(x = 1\) as undefined point.

Find the Derivative

Next, find the derivative of the function. Using the Quotient Rule, the derivative \(y'=\frac{\ln x-1}{(\ln x)^2}\) is obtained.

Find the Relative Extrema

Then, solve for \(x\) when \(y' = 0\). This gives \(x = e\). Substitute \(x = e\) into the original equation to obtain \(y = e\). Therefore, there is a relative maximum at \(x = e\).

Find the Second Derivative

Find the second derivative, \(y''\), of the function. This gives \(y'' = \frac{-1}{x (\ln x)^3}\).

Find Points of Inflection

Find the points of inflection by setting \(y'' = 0\), which gives no solution. Hence, there are no points of inflection for this function.

Verification Using a Graphing Utility

Finally, using a graphing utility, plot the function \(y=\frac{x}{\ln x}\) to verify the findings. There should be a relative maximum at \(x = e\) and there should not be any points of inflection.

Key concepts

Relative Extrema

Understanding the concept of relative extrema is a vital part of analyzing the behavior of functions, especially when graphing rational functions like \(y=\frac{x}{\ln x}\). In calculus, relative extrema refer to the points on a graph where the function reaches a local maximum or minimum. These points are determined by analyzing the first derivative of the function.

Finding the first derivative using derivative calculus and setting it equal to zero can reveal the potential relative extrema. For instance, the derivative \(y'=\frac{\ln x-1}{(\ln x)^2}\) obtained from our function indicates where the slope of the tangent line is zero, and potentially where relative maxima or minima occur. When solving for \(x\) in \(y' = 0\), we found that \(x = e\) gives us a critical point. By substituting back into the original formula, we establish that there is a relative maximum at \(x = e\), with the corresponding y-value being \(e\).

Remember, just finding the critical points doesn't guarantee a maximum or minimum; it's also necessary to test these points. A common method is the first derivative test, which involves analyzing the sign of the derivative before and after the critical point. In this exercise, we found a relative maximum at \(x = e\) based on this procedure.

Points of Inflection

Points of inflection are another intriguing aspect of graphing rational functions. They are the points where the curvature changes from concave up (like a cup) to concave down (like a cap), or vice versa. To identify these points, we often use the second derivative of the function.

In our example, by finding the second derivative \(y'' = \frac{-1}{x (\ln x)^3}\), we sought to determine where the concavity of the function changes. The points of inflection occur where \(y'' = 0\) or where the second derivative is undefined. However, our exercise presents a unique case where the second derivative does not equal zero for any value of \(x\), which suggests that there are no points of inflection for this specific function.

It's crucial to note that the absence of points of inflection does not imply that the function lacks complexity or interesting features. It solely indicates that the curvature of the function does not change direction. Verification with a graphing utility would show a smooth curve without any inflection points for the function \(y=\frac{x}{\ln x}\).

Derivative Calculus

Derivative calculus, often simply called 'derivatives', is the mathematical study of how functions change — it is fundamental in finding relative extrema, points of inflection, and analyzing the overall behavior of functions. For graphing a rational function such as \(y=\frac{x}{\ln x}\), derivatives help us understand the function's rate of change at any point.

The first derivative, for example, reveals the slope of a function at a given point and is pivotal in locating relative extrema. In the given exercise, we used the Quotient Rule — a technique for finding the derivative of a function that is the ratio of two functions — to determine the first derivative. Similarly, the second derivative, which provides information on the function's curvature, aids us in detecting points of inflection.

Understanding how to calculate and interpret these derivatives is essential for students to effectively graph and analyze functions. The exercise effectively demonstrates how derivative calculus is applied to unravel the characteristics of a rational function, illustrating the importance of this branch of mathematics in real-world problem-solving.