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. $$ y=\frac{\ln x}{x} $$

Short answer

The function \(y=\frac{\ln x}{x}\) has a relative extrema at \(x=1\) and an inflection point at \(x=1/2\). The graph of the function confirms these points.

Step-by-step solution

Understanding the Function

The function provided \(y = \frac{\ln x}{x}\) is a combination of the logarithmic function (natural log in this case) and a rational function. The domain of this function is \(x > 0\), since the logarithm is undefined for \(x \leq 0\) and division by zero is undefined.

Find the Derivative

To analyze the function, the first and second derivatives are required. The first derivative can be found using the quotient rule: If \(\frac{f}{g}\) are differentiable functions, then the derivative of \(\frac{f}{g}\) is given by \(\frac{g(f')-g'(f)}{g^2}\). In our case, \(f(x) = \ln x\) and \(g(x) = x\). Applying a quotient rule: \(y' = \frac{x(1/x) - 1*\frac{1}{x}}{x^2} = \frac{1 - 1/x }{x}\). Simplifying that gives us \(y' = \frac{x - 1}{x^2}\) as the first derivative.

Find the Relative Extrema

The relative extrema are found by setting the first derivative to zero and solving for x. i.e. \(y' = 0 \rightarrow \frac{x - 1}{x^2} = 0 \rightarrow x = 1\). Therefore at \(x=1\), we have a relative extrema.

Find the Inflection Points

The inflection points are found by setting the second derivative to zero and solving for x. First, find the second derivative \(y''\), then set \(y'' = 0\) and solve for x. \(y'' = (2x - 1) / (x^3)\). Set \(y'' = 0 \rightarrow \frac{2x - 1}{x^3} = 0 \rightarrow x = 1/2.\) Therefore, at \(x=1/2\) we have a point of inflection.

Use a Graphing Utility to Verify

Using a graphing calculator or an online graphing utility, plot \(y = \frac{\ln x}{x}\). The relative extrema at \(x=1\) and the inflection point at \(x=1/2\) should coincide with the calculated points, verifying your results.

Key concepts

Relative Extrema

When graphing functions, relative extrema are points where the function reaches its minimum or maximum values within a certain interval. These points are crucial as they illustrate where a function's growth changes direction. In the problem, to find the relative extrema of the function \(y = \frac{\ln x}{x}\), we started by finding its first derivative. By setting the first derivative \(y' = \frac{x - 1}{x^2}\) to zero, we discovered that the function has a relative extremum at \(x=1\). This means at \(x=1\), the function either reaches a peak (maximum) or a valley (minimum) in its curvature before changing direction.

Points of Inflection

Points of inflection are places on the graph where the curvature changes sign. This tells us how the function is bending. When a function goes from being concave up (like a cup) to concave down (like a frown), or the other way around, it's at a point of inflection. For our function \(y = \frac{\ln x}{x}\), after finding the second derivative \(y'' = \frac{2x - 1}{x^3}\), we set it to zero to find such points. Solving \(y'' = 0\) gives us \(x = 1/2\), indicating a point of inflection at this x-value. This helps visualize how steep or flat sections of a function's graph will transition across different ranges of x.

Domain of a Function

Understanding the domain of a function is fundamental. It's all about pinpointing which x-values you can plug into the function without breaking mathematical rules. For \(y = \frac{\ln x}{x}\), the function has a domain of \(x > 0\). This is because the logarithmic function \(\ln x\) is only defined for positive values of x, and we must avoid division by zero. The domain clearly tells us which segments of the x-axis are usable for our function, ensuring that our graph is plotted over valid inputs only.

Quotient Rule

The quotient rule is a handy tool in calculus for finding the derivatives of functions that are divided by each other. It comes into play often when analyzing functions like \(y = \frac{\ln x}{x}\). The rule states that if you have a function \(\frac{f}{g}\), its derivative is \(\frac{g \cdot f' - f \cdot g'}{g^2}\). In our problem, with \(f(x) = \ln x\) and \(g(x) = x\), applying the quotient rule allowed us to find the first derivative \(y' = \frac{x - 1}{x^2}\). This step is vital for understanding how changes in x will affect y, providing insight into slope and potential extrema.

Graphing Calculator

A graphing calculator is an essential tool for visualizing functions and verifying calculus results. By inputting \(y = \frac{\ln x}{x}\) into a graphing calculator, you can see a visual representation of everything calculated manually, like relative extrema and points of inflection. It allows you to quickly check if the relative minimum or maximum at \(x=1\) and the inflection at \(x=1/2\) match your calculations. Using graphing technology enhances understanding by linking numerical results to their graphical counterparts, imparting a deeper grasp of concepts in calculus.