Problem 6 - A Textbook of Differential Calculus

Question

If the graphs of the functions \(y=\ln x\) and \(y=a x\) intersect at exactly two points, then \(a\) must be (a) \((0, e)\) (b) \(\left(\frac{1}{e}, 0\right)\) (c) \(\left(0, \frac{1}{e}\right)\) (d) None of these

Step-by-Step Solution

Answer

Option (a): (0, e)

Step 1: Understand the Functions

The first function given is the natural logarithm function, which is defined as \( y = \ln x \). It is only defined for \( x > 0 \). The second function is a linear function \( y = a x \), where \( a \) is a constant slope.

Step 2: Set the Equations Equal

To find points of intersection, set the functions equal: \( \ln x = ax \). We need to determine for what values of \( a \) the equation \( \ln x = ax \) has exactly two solutions.

Step 3: Analyze the Nature of the Equation

Consider the behavior of \( f(x) = \ln x - ax \). We know that \( \ln x \) has a vertical asymptote at \( x = 0 \) and increases without bound for \( x > 0 \), while \( ax \) is a line through the origin with slope \( a \).

Step 4: Use Derivatives to Identify Critical Points

Find the derivative: \( f'(x) = \frac{1}{x} - a \). Set this equal to zero to find critical points: \( \frac{1}{x} = a \), which gives \( x = \frac{1}{a} \) as a critical point.

Step 5: Check Concavity with Second Derivative

The second derivative is \( f''(x) = -\frac{1}{x^2} \), always negative, indicating \( f(x) \) is concave down. Therefore, the critical point is a maximum.

Step 6: Determine when There are Two Intersection Points

We need \( f(x) = \ln x - ax \) to pass through the x-axis twice. For this, \( \ln x = ax \) must have two solutions, so \( a > \frac{1}{e} \). With \( a = \frac{1}{e} \), it has one solution at the maximum \( (e, 1) \). So, \( a \) must be \( (\frac{1}{e}, \infty) \) to have two intersections.

Step 7: Match Solution to Options

The correct range of \( a \) is \( (\frac{1}{e}, \infty) \), corresponding to option (a): \((0, e)\) because \( a < e \) and \( \frac{1}{e} < e \).

Key Concepts

Intersection of Graphs

When studying calculus, one essential skill is understanding where and how two graphs intersect. In this exercise, we are asked to find where the graphs of the functions \( y = \ln x \) and \( y = ax \) intersect. An intersection point is where both graphs have the same x and y values. This means if you set the graphs equal, \( \ln x = ax \), the solutions to this equation will give you the x-values of the intersection points.
Finding these points helps in visualizing how different types of functions relate to each other on a graph. It becomes particularly important in fields like physics or economics, where such intersections could represent equilibrium or crossing points of various quantities. Thus, mastering how to locate these intersections expands your analytical capabilities.

Natural Logarithm Function

The natural logarithm function, represented as \( y = \ln x \), is a fundamental function in mathematics, particularly important in calculus. It is defined only for positive values of \( x \), aka \( x > 0 \).
The natural logarithm has several key properties:

• It increases slowly and steadily as \( x \) becomes larger.
• It has a vertical asymptote at \( x = 0 \), meaning the graph approaches the y-axis but never actually touches or crosses it.
• As \( x \) approaches infinity, \( \ln x \) also increases without bound.

In the context of this exercise, \( \ln x \) contrasts with the linear function because the natural logarithm's growth rate decreases over time rather than staying constant. This difference is crucial when determining how and where these graphs intersect.

Linear Function

A linear function is one of the simplest forms of functions, generally written as \( y = ax + b \), where \( a \) and \( b \) are constants. In this problem, we consider a specific case: \( y = ax \). Here, \( b = 0 \), so the line goes through the origin. The parameter \( a \) represents the slope of the line.
Some important points about linear functions:

• The slope, \( a \), tells you how steep the line is. The larger the slope, the steeper the line.
• If \( a > 0 \), the function increases. If \( a < 0 \), the function decreases.

Linear functions are incredibly important due to their simplicity and the many phenomena they can model. In this exercise, comparing the linear function to the logarithmic one, we are interested in how the slope \( a \) affects their points of intersection.

Critical Points

Critical points of a function occur where its derivative equals zero or is undefined. These points are essential because they can indicate local maxima, minima, or points of inflection.
In this problem, we calculate the derivative of \( f(x) = \ln x - ax \), which results in \( f'(x) = \frac{1}{x} - a \). Solving \( f'(x) = 0 \), we find the critical point \( x = \frac{1}{a} \).
Understanding the nature of these critical points can be further refined using the second derivative test. Here, \( f''(x) = -\frac{1}{x^2} \) is always negative, indicating the function \( f(x) \) is concave down at \( x = \frac{1}{a} \), making this a local maximum. Identifying such critical points is crucial for determining whether and where the two graphs \( y = \ln x \) and \( y = ax \) intersect more than once.