Difference between revisions of "Trivial Inequality"
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# (''Introductory'') Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=4xy+6yz+2z-1</math>. | # (''Introductory'') 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=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=9</math> and <math>BC: AC=40: 41</math>. What is the largest area that this triangle can have? | ||
− | # ([[1969 Canadian MO Problems/Problem 3]]) Let <math>c</math> be the length of the [[hypotenuse]] of a [[right triangle]] whose two other sides have lengths <math>a</math> and <math>b</math>. Prove that <math>a+b\le c\sqrt{2}</math>. When does the equality hold? | + | # ([[1969 Canadian MO Problems/Problem 3|1969 Canadian MO]]) Let <math>c</math> be the length of the [[hypotenuse]] of a [[right triangle]] whose two other sides have lengths <math>a</math> and <math>b</math>. Prove that <math>a+b\le c\sqrt{2}</math>. When does the equality hold? |
[[Category:Inequality]] | [[Category:Inequality]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 07:56, 17 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 ,
, with equality if and only if
.
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
- (Introductory) Find all integer solutions
of the equation
.
- (AIME 1992) Triangle
has
and
. What is the largest area that this triangle can have?
- (1969 Canadian MO) Let
be the length of the hypotenuse of a right triangle whose two other sides have lengths
and
. Prove that
. When does the equality hold?