Problem 26 - Calculus for Biology and Medicine

Question

Show algebraically that if \(n \geq m, x^{n} \leq x^{m}\), for \(0 \leq x \leq 1\), and \(x^{n} \geq x^{m}\), for \(x \geq 1 .\)

Step-by-Step Solution

Answer

For \(n \geq m\), \(x^{n} \leq x^{m}\) if \(0 \leq x \leq 1\) and \(x^{n} \geq x^{m}\) if \(x \geq 1\).

Step 1: Analyze the Condition for \(0 \leq x \leq 1\)

In this range, since \(x\) is less than or equal to 1, raising \(x\) to a higher power, meaning using a larger exponent \(n\) compared to \(m\), results in a smaller value. This is because \(x^{n} = x^{m} \cdot x^{n-m}\) and since \(x \leq 1\), \(x^{n-m} \leq 1\). Therefore, \(x^{n} \leq x^{m}\) is true when \(n \geq m\).

Step 2: Analyze the Condition for \(x \geq 1\)

For \(x \geq 1\), any positive number raised to a higher power will yield a larger value. We have \(x^{n} = x^{m} \cdot x^{n-m}\) and since \(x \geq 1\), \(x^{n-m} \geq 1\). Therefore, \(x^{n} \geq x^{m}\) holds true when \(n \geq m\).

Step 3: Validate Both Conditions Algebraically

For \(x = 1\), both \(x^{n}\) and \(x^{m}\) are equal to 1, making both inequalities true. Thus, the conditions \(x^{n} \leq x^{m}\) and \(x^{n} \geq x^{m}\) are satisfied for all scenarios when \(n \geq m\).

Key Concepts

Algebraic Proof

Algebraic proof is a method used to verify mathematical statements using algebraic techniques and logical reasoning. In the given problem, we are asked to show inequalities involving powers of a number, based on the value of the base. Here's how we approach it:

• We begin by examining different cases for the base value, such as when it is within 0 to 1, and when it is equal to or greater than 1.
• We use algebraic manipulation to express the different terms of the inequality using a common base.

By using basic algebraic operations like factorization, we achieve a form where the behavior of the expression becomes evident. Then, depending on the restrictions provided for the base, the inequality can be logically verified.

Inequalities

Inequalities are expressions that compare quantities, using signs like ≤ (less than or equal to) or ≥ (greater than or equal to). In mathematical analysis, inequalities can help us understand and establish relationships between different expressions.
In this problem:

• We observe that for any base 0 ≤ x ≤ 1, the power relationship reverses due to the base being less than or equal to one.
• For base values x ≥ 1, raising the base to a higher power accentuates the size of the expression.

By understanding these relationships, we demonstrate that respective inequalities, where the power is either a dampener or an amplifier, highlight core numeric behavior, linking back to the rules of exponents.

Exponentiation

Exponentiation involves raising a base number to a power, denoted as a superscript. It represents repeated multiplication of the base number. The outcome heavily depends on both the base and the exponent:

• When 0 ≤ x ≤ 1, raising to higher powers tends to diminish value because multiplying sub-one numbers repeatedly results in smaller solutions.
• In contrast, for x ≥ 1, multiplying the base by itself increases its value exponentially with higher powers.

In other words, the magnitude of change through exponentiation depends on whether "x" lies between 0 and 1 or is greater than 1. Recognizing this allows us to predict outcomes based on these exponential behaviors.

Mathematical Analysis

Mathematical analysis provides the tools necessary to tackle complex problems by breaking them down. The analysis of exponentiation and inequalities in this exercise demonstrates how a structured approach gives insight into mathematical behavior.
Breaking the problem:

• Define and test different scenarios based on the range of x.
• Analyze the algebraic manipulations and inequalities generated.
• Conclude through logical deductions, given known properties of numbers and operations.

By examining base specific behaviors and applying logical reasoning, mathematical analysis offers a clear, reliable pathway to demonstrate theorems or statements like those provided in this problem.