Question

Find a real number \(c\) and an \(N_{0} \in \mathbb{N}\) such that \(n^{3}-3 n^{2}+4 n

Short answer

Choose \(c = 2\) and \(N_0 = 2\).

Step-by-step solution

Understanding the Problem

We need to find a real number \(c\) and a natural number \(N_0\) such that for all natural numbers \(n\) greater than or equal to \(N_0\), the inequality \(n^3 - 3n^2 + 4n < cn^3\) holds. This means that \(cn^3\) must be large enough to always be greater than \(n^3 - 3n^2 + 4n\) for sufficiently large \(n\).

Reformulating the Inequality

Rewriting the original inequality \(n^3 - 3n^2 + 4n < cn^3\), we get \(n^3 - 3n^2 + 4n - cn^3 < 0\). This simplifies to \((1-c)n^3 - 3n^2 + 4n < 0\).

Focusing on Dominating Terms

Since \(n^3\) is the highest degree term, as \(n\) becomes very large, the \(n^3\) terms will dominate. We need \((1-c)n^3\) to eventually cancel out \(-3n^2 + 4n\). Thus, \(1-c\) should be negative for the cubic term's contribution to be negative.

Choosing a Suitable \(c\)

Set \(c\) such that \(1 - c < 0\). Therefore, choose \(c = 2\). This means the inequality becomes \((1-2)n^3 - 3n^2 + 4n < 0\), i.e., \(-n^3 - 3n^2 + 4n < 0\).

Finding \(N_0\)

Examine the inequality \(-n^3 - 3n^2 + 4n < 0\). Factorize it if possible or test values:\(-n^3 - 3n^2 + 4n = -n(n^2 + 3n - 4)\). The quadratic \(n^2 + 3n - 4\) has roots (using the formula): \(n = \frac{-3 \pm \sqrt{3^2 - 4\cdot 1\cdot (-4)}}{2\cdot 1} = \{-4, 1\}\). Therefore, the quadratic is positive for \(n > 1\). Test \(n = 2\) to verify: \(-2^3 - 3(2)^2 + 4(2) = -8 - 12 + 8 = -12 < 0\). Thus, \(N_0 = 2\) works.

Key concepts

inequality

In mathematics, the concept of inequality involves comparing expressions to determine if one is greater or less than the other. In our exercise example, the inequality is stated as \(n^3 - 3n^2 + 4n < cn^3\). This system is designed to determine when one side will consistently be less than the other for certain conditions of \(n\) and \(c\).

It is essential to identify the bounds or constraints, such as real numbers like \(c\) and natural numbers greater than a certain limit \(N_0\).

• We aim to establish the situation where \(n^3 - 3n^2 + 4n\) is consistently smaller than \(cn^3\).
• The constraint holds for specific numerical conditions and larger values of \(n\).

Understanding how to manipulate and solve these inequalities allows us to predict behavior, especially as variable sizes grow indefinitely. This forms the backbone of many theoretical assessments used in mathematics and computer science.

mathematical proof

A mathematical proof involves logical reasoning to verify that a statement or theorem is consistently true under certain conditions. In proving the inequality \(n^3 - 3n^2 + 4n < cn^3\), we follow steps to break down the inequality and demonstrate its validity across defined limits.

To reformulate the given problem:

• Convert the inequality into \((1-c)n^3 - 3n^2 + 4n < 0\).
• Recognize the need for \(1-c\) to be negative to ensure the leading term \(-(n^3)\)'s substantial impact as \(n\) grows.

By choosing a suitable value for \(c\), such as \(c=2\), we simplify the proof because this ensures \((1-c) = -1\), leading to a negatively dominated cubic term.

Finding \(N_0\) is also part of this proof, ensuring that for all \(n \geq N_0\), the inequality consistently holds true. The use of factorization and root calculations further solidifies the proof ensuring logical consistency.

dominant term analysis

Dominant term analysis is a technique to solve complex expressions by focusing on the terms with the highest degree or impact. When working with polynomials, especially in asymptotic analysis, this technique is invaluable. In the inequality we were given, \(n^3\) is the dominant term as it has the highest power.

As \(n\) becomes much larger, the other terms \(-3n^2\) and \(+4n\) become less impactful compared to \(n^3\).

• The analysis involves identifying the largest contributing term to the polynomial's growth.
• With \((1-c)n^3\) contrasting with \(-3n^2 + 4n\), knowing \(1-c\) is crucial as it determines the polynomial's overall behavior efficiently as \(n\) grows larger.

By understanding which terms are most influential, we can predict which side of an inequality will dominate, guiding our choice of \(c\) and ensuring the inequality holds at sufficiently large \(n\). This method simplifies otherwise complex evaluations and supports clear mathematical and computational reasoning.