Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) The inequality \(2^{x} \geq x+1\) is true for all positive real numbers \(x\).

Short answer

The inequality \(2^{x} \geq x+1\) is true for all positive real numbers \(x\). This is confirmed by checking the base case, graphing the functions, and comparing their derivatives. In conclusion, the inequality does hold true.

Step-by-step solution

Check for base cases

We check the base case, AS \(x=1\), we have \(2^{1}=1+1\), which is true. Therefore, the inequality holds for \(x=1\).

Draw the functions

Now, let's graph the functions \(y=2^{x}\) and \(y=x+1\). Using graphing aids, it can be shown that for positive \(x\), the graph of \(2^{x}\) is always above the line \(y=x+1\).

Investigate Derivatives

To confirm this, we can look at the derivative of both functions. The derivative of \(2^{x}\) is \(2^{x} \ln(2)\), and the derivative of \(x+1\) is \(1\). When \(x>0\), it's clear that \(2^{x} \ln(2) > 1\). So, the exponential function grows faster than the linear function for all \(x>1\). Therefore, we have confirmed that the inequality \(2^{x} \geq x+1\) holds for all positive real numbers \(x\).

Consider \(x=0\)

As a special case, for \(x=0\), we have \(2^{0}=1\) and \(0+1=1\), so the inequality also holds for \(x=0\).

Key concepts

Understanding Inequalities

Inequalities are mathematical expressions that relate two values, showing that one is greater than, less than, or equal to the other. The inequality in the exercise provided, \(2^{x} \geq x+1\), is an example of a relational statement, where we're comparing the exponential function \(2^{x}\) to the linear function \(x+1\) for all positive real numbers \(x\).

To determine whether such inequalities hold true for certain values of \(x\), we can use various methods, including but not limited to checking specific cases, drawing graphs, or investigating the behaviors of the functions through their derivatives. When proving inequalities, it is critical to scrutinize the behavior over the entire range of interest to ensure the statement is universally true.

Exponential Functions and Their Growth

Exponential functions, like \(2^{x}\), are known for their rapid growth rates. These functions have the form \(a^{x}\), where \(a\) is a constant, and \(x\) represents the exponent. As \(x\) increases, exponential functions grow much faster than polynomial functions, such as the linear function \(x+1\) used in our comparison.

When comparing the behavior of \(2^{x}\) and \(x+1\), it's apparent in the graphing stage that for positive \(x\), \(2^{x}\) surges ahead and surpasses the linear function quite dramatically as \(x\) gets larger. This visual analysis is a powerful tool in establishing the truth of the given inequality over a general set of values.

Delving into Derivatives

Derivatives are a cornerstone of calculus, providing insights into the rate of change of functions. When comparing \(2^{x}\) and \(x+1\), we look at their derivatives to understand how their rate of change differs for positive \(x\). The derivative of \(2^{x}\) with respect to \(x\) is \(2^{x} \ln(2)\), indicating that as \(x\) increases, the rate at which \(2^{x}\) grows is itself an exponentially increasing function. In contrast, the derivative of the linear function \(x+1\) is a constant \(1\).

This disparity in growth rates is crucial. The derivative of \(2^{x}\) not only starts out larger than that of \(x+1\) for any \(x>0\), but it also accelerates away from the constant growth of the linear function. This mathematical fact supports our proof that the inequality \(2^{x} \geq x+1\) holds true for all positive real numbers \(x\), ensuring a deeper and holistic understanding of the concept.