How can I use theorems to derive a statement about inequalities?

  • Context: Undergrad 
  • Thread starter Thread starter evagelos
  • Start date Start date
  • Tags Tags
    deriving
Click For Summary
SUMMARY

The discussion focuses on deriving inequalities using established mathematical theorems, specifically the AM-GM inequality and identities related to squares of differences. The inequalities presented include \(a^2 + b^2 + c^2 \geq ab + ac + bc\) and \((a-b)^2 \geq 0\), \((b-c)^2 \geq 0\), \((a-c)^2 \geq 0\). Participants emphasize the importance of applying these identities systematically to validate the derived statements.

PREREQUISITES
  • Understanding of the AM-GM inequality
  • Familiarity with algebraic identities involving squares
  • Basic knowledge of inequalities in mathematics
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study the derivation of the AM-GM inequality in detail
  • Practice proving inequalities using algebraic identities
  • Explore advanced topics in inequality theory
  • Learn about other inequalities such as Cauchy-Schwarz and Jensen's inequality
USEFUL FOR

Mathematics students, educators, and anyone interested in deepening their understanding of inequalities and their applications in various mathematical contexts.

evagelos
Messages
314
Reaction score
0
By using appropriate theorems derive the following statement :

[tex]a^2 + b^2 + c^2\geq ab + bc +ac[/tex]
 
Mathematics news on Phys.org
[tex](a-b)^2 \geq 0[/tex]
[tex](b-c)^2 \geq 0[/tex]
[tex](a-c)^2 \geq 0[/tex]
Try using these identities to derive your statement. Or if you've learn about the AM-GM inequality, then use it one two variables on the right at a time (this is the same as using the identities above).
 

Similar threads

Undergrad Are there any two pairs of integers with the same result in a specific function?
  • · Replies 7 ·
Replies
7
Views
2K
MHB How do I calculate the side of a rhombus using the bisector of an angle theorem?
  • · Replies 1 ·
Replies
1
Views
2K
MHB High school inequality 5 abc+acb+bca≥a+b+c.
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad A question about Young's inequality and complex numbers
  • · Replies 13 ·
Replies
13
Views
2K
Equality of sums of powers
  • · Replies 47 ·
2
Replies
47
Views
7K
MHB Therefore, AB + BC = AC, thus proving that the given equation is true.
  • · Replies 3 ·
Replies
3
Views
6K
Undergrad Proof Using Rearrangement Inequality
  • · Replies 2 ·
Replies
2
Views
2K
MHB Proving Miquel's Theorem: Need Help!
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What is the inequality for prime numbers in the Prime Number Theorem proof?
  • · Replies 9 ·
Replies
9
Views
2K
Use Remainder theorem to find factors of ##(a-b)^3+(b-c)^3+(c-a)^3##
  • · Replies 4 ·
Replies
4
Views
2K