Difference between revisions of "Trivial Inequality"
(proof) |
(→Proof: three) |
||
Line 5: | Line 5: | ||
==Proof== | ==Proof== | ||
− | We assume the negation of the theorem; that is there is a real <math>x</math> such that <math>x^2<0</math>. We now consider | + | We assume the negation of the theorem; that is there is a real <math>x</math> such that <math>x^2<0</math>. We now consider three cases: |
'''Case 1; <math>x=0</math>:''' This obviously is a contradiction, as <math>0^2\not < 0</math> | '''Case 1; <math>x=0</math>:''' This obviously is a contradiction, as <math>0^2\not < 0</math> |
Revision as of 21:40, 22 November 2007
The Trivial Inequality is a simple inequality named because of its sheer simplicity and seeming triviality.
Contents
Inequality
The inequality states that for all real numbers
. This is a rather useful inequality for proving that certain quantities are nonnegative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique.
Proof
We assume the negation of the theorem; that is there is a real such that
. We now consider three cases:
Case 1; : This obviously is a contradiction, as
Case 2; : Here, we divide by
, which is allowable because we know
is positive:
, which results in contradiction.
Case 3; : Since
, we can again divide by
and reverse the inequality symbol:
, which again is a contradiction.
Thus, the theorem is true.
Applications
The trivial inequality can be used to maximize and minimize quadratic functions.
After completing the square, the trivial inequality can be applied to determine the extrema of a quadratic function.
Here is an example of the important use of this inequality:
Suppose that are nonnegative real numbers. Starting with
, after squaring we have
. Now add
to both sides of the inequality to get
. If we take the square root of both sides (since both sides are nonnegative) and divide by 2, we have the well-known Arithmetic Mean-Geometric Mean Inequality for 2 variables:
Problems
Introductory
- Find all integer solutions
of the equation
. (No source nor solution)
Intermediate
- Triangle
has
and
. What is the largest area that this triangle can have? (Source)