Difference between revisions of "Trivial Inequality"
(→See also) |
m (→Problems) |
||
Line 19: | Line 19: | ||
== Problems == | == Problems == | ||
− | # Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2= | + | # Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=4xy+6yz+2z-1</math>. |
# ([[1992 AIME Problems/Problem 13|AIME 1992]]) Triangle <math>ABC</math> has <math>AB</math><math>=9</math> and <math>BC: AC=40: 41</math>. What is the largest area that this triangle can have? | # ([[1992 AIME Problems/Problem 13|AIME 1992]]) Triangle <math>ABC</math> has <math>AB</math><math>=9</math> and <math>BC: AC=40: 41</math>. What is the largest area that this triangle can have? | ||
[[Category:Inequality]] | [[Category:Inequality]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 02:47, 15 March 2008
The trivial inequality is an inequality that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness.
Contents
Statement
For all real numbers , .
Proof
We proceed by contradiction. Suppose there exists a real such that . We can have either , , or . If , then there is a clear contradiction, as . If , then gives upon division by (which is positive), so this case also leads to a contradiction. Finally, if , then gives upon division by (which is negative), and yet again we have a contradiction.
Therefore, for all real , as claimed.
Applications
The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
One application is maximizing and minimizing quadratic functions. It gives an easy proof of the two-variable case of the Arithmetic Mean-Geometric Mean inequality:
Suppose that and are nonnegative reals. By the trivial inequality, we have , or . Adding to both sides, we get . Since both sides of the inequality are nonnegative, it is equivalent to , and thus we have as desired.
Problems
- Find all integer solutions of the equation .
- (AIME 1992) Triangle has and . What is the largest area that this triangle can have?