Proving Inequalities Involving Positive Real Numbers

  • Thread starter Sam Morse
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In summary, the conversation is about proving the inequality a^3+b^3+c^3≥a^2b+b^2c+c^2a for positive real numbers a, b, and c. The attempt at a solution involves assuming a≥b≥c>0 and shifting the left side of the inequality to the right side. The question of how to factorize the expression is raised, and the conversation discusses possible approaches. A messy but possible solution is suggested, but then a factorization is found on a website which leads to the belief that the inequality is true.
  • #1
Sam Morse
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Homework Statement



Let a,b,c be positive real numbers. Prove that:

a3+b3+c3≥a2b+b2c+c2a

Homework Equations





The Attempt at a Solution



I assumed that a≥b≥c>0 following which I shifted the left side of this inequality to the right side giving

a3+b3+c3-(a2b+b2c+c2a)≥0

How do I do the required factorisation ... ?
 
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  • #2
Are you sure that there is an easy way to factorize this?

I would expect that this can be reduced to a problem similar to a/b + b/c + c/a > 3.
 
  • #3
I see a way to solve it, though it is a bit messy. First divide the entire equation by abc .
A bit of algebra gets [itex] a^3+b^3+c^3 \geq ab + bc+ ac [/itex]. We can assume that a,b, and c are all greater than 1 since if (a,b,c) satisfies the inequality, then so does (ax,bx,cx) for any positive x. Using the fact that [itex] x^3 \geq x^2 if x>1 [/itex], it is therefore sufficient to show that
[itex] a^2+b^2+c^2 \geq ab + bc +ac [/itex]

Get everything over to the left hand side and multiply by 2. You will see that it factors nicely.

Edit: Oops, disregard what I said, I made a mistake in my algebra.
 
Last edited:
  • #4

1. What is factorisation and why is it important?

Factorisation is the process of breaking down a mathematical expression into its smaller factors or components. It is important because it allows us to simplify complex expressions, solve equations, and find common factors in different mathematical concepts.

2. What are some common challenges or problems with factorisation?

One of the most common problems with factorisation is identifying the correct method to use for a given expression. There are multiple methods, such as GCF (greatest common factor), difference of squares, and trinomial factoring, which can be confusing. Another challenge is knowing when to stop factoring and if there are any remaining factors left.

3. How can I improve my factorisation skills?

Practicing regularly and reviewing different methods for factoring can improve your skills. It is also helpful to familiarize yourself with common patterns and techniques for different types of expressions. Additionally, breaking down larger expressions into smaller, more manageable parts can make the process easier.

4. Can factorisation be used in real-life situations?

Yes, factorisation has real-world applications in various fields such as finance, engineering, and computer science. It can be used to simplify complex equations, find the most efficient way to allocate resources, and optimize algorithms.

5. Are there any alternative methods to factorisation?

Yes, there are alternative methods for simplifying expressions, such as using a calculator or computer software. However, understanding the principles and techniques of factorisation can help in solving problems and understanding mathematical concepts more deeply.

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