Problem 69 - Precalculus with Limits

Question

In Exercises 63-70, use the function and its derivative to determine any points on the graph of \(f\) at which the tangent line is horizontal. Use a graphing utility to verify your results. \(f(x) = x\ \textrm{ln}\ x, \quad f'(x) = \textrm{ln}\ x + 1\)

Step-by-Step Solution

Answer

The point on the graph of the function \(f(x) = x\ \textrm{ln}\ x\) where the tangent line is horizontal is \((e^{-1}, -e^{-1})\).

Step 1: Set derivative equal to zero

Set the derivative \(f'(x)\) equal to zero, as we are looking for points where the slope of the tangent line, which is represented by the derivative, is zero. This will give us: \(\textrm{ln}\ x + 1 = 0\).

Step 2: Solve for x

In order to find the x-values of these points, we need to solve the equation from step 1 for \(x\). This can be done by subtracting 1 from both sides of the equation, which gives us: \(\textrm{ln}\ x = -1\). Then we can find the exponent by using the property of natural logarithms that states \(\textrm{ln}\ a = b\) is equivalent to \(e^b = a\). So by applying this property, we find that \(x = e^{-1}\).

Step 3: Substitute x value in the function

Substitute the x value obtained in the previous step into the function to obtain the corresponding y value. The point \(P\) where the tangent line is horizontal is \((e^{-1}, e^{-1} \textrm{ln}(e^{-1}))\). Using the property \(\textrm{ln}\ a^{n} = n \textrm{ln}\ a\), we find the y-coordinate of the point \(P\) to be \(e^{-1}(-1) = -e^{-1}\). Therefore, the point \(P\) is \((e^{-1}, -e^{-1})\).

Step 4: Verify

To confirm the results, we can use a graphing utility to graph both the function \(f(x) = x\ \textrm{ln}\ x\) and the line \(y = -e^{-1}\), which is horizontal. At the intersection point of the graph with the horizontal line, the tangent is horizontal. Thus this confirms that \((e^{-1}, -e^{-1})\) is the point on the function where the tangent is horizontal.

Key Concepts

Derivative

The concept of a derivative is fundamental in calculus and describes how a function changes as its input changes. Specifically, the derivative of a function at a certain point gives the slope of the tangent line to the graph of the function at that point.

• If the slope is positive, the function is increasing at that point.
• If it is negative, the function is decreasing.
• If the derivative is zero, the function has a horizontal tangent line.

When dealing with the example function, \(f(x) = x\ \ln\ x\), we calculated the derivative as \(f'(x) = \ln\ x + 1\). Understanding this, we can identify the behavior of \(f(x)\) by setting its derivative to zero. Solving \(\ln\ x + 1 = 0\) led us to the point \(x = e^{-1}\), indicating where the slope of the tangent is zero, and thus horizontal.

Natural Logarithm

The natural logarithm, often written as \(\ln x\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. Here are some important properties of natural logarithms to keep in mind:

• Natural logarithms are only defined for positive real numbers.
• The expression \(\ln(ab) = \ln a + \ln b\).
• The logarithm of a power, \(\ln(a^b) = b\ln a\).

In our exercise, the function \(f(x) = x \ln x\) uses the natural logarithm. To find where the tangent is horizontal, we solved the equation \(\ln x = -1\). Using the property \(\ln a = b\) equivalent to \(e^b = a\), we deduced that \(x = e^{-1}\). This illustrates how logarithmic properties come into play during problem-solving.

Graphing Utility

A graphing utility is a tool, usually a calculator or software, that helps visualize functions by plotting them on a coordinate plane. This visualization can confirm mathematical results by showing the graph's behavior and its relationship with lines such as the horizontal tangent.Here’s how you can use a graphing utility effectively:

• Input the function you want to study, in our example \(f(x) = x \ln x\).
• Plot any lines of interest, such as horizontal lines at critical points you computed, like \(y = -e^{-1}\).
• Check the graph to verify intersections, confirming where the tangent line is horizontal.

By using a graphing utility, we graphically verified that the point \((e^{-1}, -e^{-1})\) truly shows a horizontal tangent, confirming our calculations with a visual representation. This aligns mathematical theory with practice, ensuring accuracy and a deeper understanding of the function's behavior.