Problem 70 - Thomas Calculus

Question

a. Prove that \(f(x)=x-\ln x\) is increasing for \(x > 1\) b. Using part ( a), show that \(\ln x < x\) if \(x > 1\)

Step-by-Step Solution

Answer

a. \(f(x)\) is increasing for \(x>1\); b. \(\ln x < x\) for \(x>1\).

Step 1: Define the Derivative

First, we need to find the first derivative of the function \(f(x) = x - \ln x\). The derivative of \(x\) is 1, and the derivative of \(\ln x\) is \(\frac{1}{x}\). Thus, the first derivative \(f'(x)\) is: \[ f'(x) = 1 - \frac{1}{x} \]

Step 2: Determine where the Derivative is Positive

The function \(f(x)\) is increasing if its derivative \(f'(x)\) is greater than 0. Therefore, we solve: \[ 1 - \frac{1}{x} > 0 \] Rearranging the inequality gives: \[ 1 > \frac{1}{x} \] \[ x > 1 \] Thus, \(f(x) = x - \ln x\) is increasing for \(x > 1\).

Step 3: Relate to the Given Inequality

Since \(f(x) = x - \ln x\) is increasing for \(x > 1\), and knowing that at \(x = 1\), \(f(1) = 1 - \ln 1 = 1\), it follows that for \(x > 1\), \(f(x) \geq f(1) = 0\). This can be written as: \( x - \ln x \geq 0 \).

Step 4: Conclude the Solution

From \(x - \ln x \geq 0\), we can isolate \(\ln x\) to get \(\ln x \leq x\). Since we have established \(f(x)\) is strictly increasing, it follows that for \(x > 1\), \(\ln x < x\).

Key Concepts

Derivative

The derivative is a fundamental concept in calculus. It measures how a function changes as its input changes. Essentially, it represents the function's rate of change or the slope of its graph at any given point.
For the function \( f(x) = x - \ln x \), finding the derivative tells us how \( f(x) \) behaves for different values of \( x \). Here's how we derive it:

• The derivative of \( x \) with respect to \( x \) is \( 1 \).
• The derivative of the natural logarithm, \( \ln x \), is \( \frac{1}{x} \).

By applying these rules, the derivative of \( f(x) \) is given by:\[ f'(x) = 1 - \frac{1}{x} \]Understanding this derivation is crucial, especially when analyzing the increasing or decreasing nature of functions, as it provides a quantitative measure of the function's behavior.

Increasing Function

An increasing function is one that consistently grows with the increase in its input value. For a function \( f(x) \), it is considered increasing if, for any two points \( a \) and \( b \), where \( a < b \), we have \( f(a) < f(b) \).
To determine if a function is increasing, we examine its first derivative. For example, if the derivative \( f'(x) \) is greater than zero for all \( x \) in some interval, the function is increasing on that interval.

• From the calculation, \( f'(x) = 1 - \frac{1}{x} \).
• Setting the inequality \( 1 - \frac{1}{x} > 0 \) leads to solving \( x > 1 \).

This analysis confirms that \( f(x) = x - \ln x \) is increasing for \( x > 1 \). This property is used to further investigate the inequality involving \( \ln x \).

Inequality

Inequalities express the relative size or order of two values. They are fundamental in solving and analyzing various mathematical scenarios. For this function, we want to show \( \ln x < x \) when \( x > 1 \).
Using the property of increasing functions:

• Since \( f(x) = x - \ln x \) is increasing for \( x > 1 \), and \( f(1) = 1 - \ln 1 = 1 \).
• This suggests for \( x > 1 \), \( f(x) \geq f(1) = 0 \).

We express this information as \( x - \ln x \geq 0 \), which simplifies to \( \ln x \leq x \). Because the function is strictly increasing, we conclude \( \ln x < x \) for all \( x > 1 \). Understanding inequalities in this context is crucial for determining the behavior and relationships within mathematical functions.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. It has unique properties that make it very useful in calculus and various applications.
One key property is its behavior under differentiation. The derivative of \( \ln x \) is \( \frac{1}{x} \), which is crucial when analyzing functions involving logarithms.
For the function \( f(x) = x - \ln x \), understanding the interplay between \( x \) and \( \ln x \) helps in establishing properties like increasing behavior and inequalities. We observe how \( \ln x \) compares to \( x \), especially in the context of growth. This is evidenced by showing \( \ln x < x \) for \( x > 1 \) through derivative analysis.
The natural logarithm frequently appears in problems involving growth processes, decay, and is also central in calculus transformations.