Difference between revisions of "Trivial Inequality"
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After [[completing the square]], the trivial inequality can be applied to determine the extrema of a quadratic function. | After [[completing the square]], the trivial inequality can be applied to determine the extrema of a quadratic function. | ||
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+ | Here is an example of the important use of this inequality: | ||
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+ | Suppose that <math>a,b</math> are nonnegative [[real number]]s. Starting with <math>(a-b)^2\geq0</math>, after squaring we have <math>a^2-2ab+b^2\geq0</math>. Now add <math>4ab</math> to both sides of the inequality to get <math>a^2+2ab+b^2=(a+b)^2\geq4ab</math>. If we take the square root of both sides (since both sides are nonnegative) and divide by 2, we have the well-known [[AMGM | Arithmetic Mean-Geometric Mean]] Inequality for 2 variables: <math>\frac{a+b}2\geq\sqrt{ab}</math> | ||
== Problems == | == Problems == |
Revision as of 13:26, 15 November 2007
The Trivial 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.
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
Intermediate
- Triangle
has
and
. What is the largest area that this triangle can have?
- Solution: First, consider the triangle in a coordinate system with vertices at
,
, and
.
Applying the distance formula, we see that.
- We want to maximize
, the height, with
being the base. Simplifying gives
. To maximize
, we want to maximize
. So if we can write:
then
is the maximum value for
. This follows directly from the trivial inequality, because if
then plugging in
for
gives us
. So we can keep increasing the left hand side of our earlier equation until
. We can factor
into
. We find
, and plug into
. Thus, the area is
.
- Solution: First, consider the triangle in a coordinate system with vertices at