Trivial Inequality
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
[hide]Statement
For all real numbers ,
.
Proof
We can have either ,
, or
. If
, then
. If
, then
by the closure of the set of positive numbers under multiplication. Finally, if
, then
again by the closure of the set of positive numbers under multiplication.
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.
Another application will be to minimize/maximize quadratics. For example,
Then, we use trivial inequality to get if
is positive and
if
is negative.
Problems
Introductory
- Find all integer solutions
of the equation
.
- Show that
. Solution
- Show that
for all real
and
.
Intermediate
- Triangle
has
and
. What is the largest area that this triangle can have? (AIME 1992)
- The fraction,
where and
are side lengths of a triangle, lies in the interval
, where
and
are rational numbers. Then,
can be expressed as
, where
and
are relatively prime positive integers. Find
. (Solution here see problem 3 solution 1)
Olympiad
- 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? (1969 Canadian MO)
- Let
and
be real numbers. Show that
(Solution here see problem 13 solution 1)