Difference between revisions of "Trivial Inequality"
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The trivial inequality states that <math>{x^2 \ge 0}</math> for all x. This is a rather useful inequality for proving that certain quantities are non-negative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique. | The trivial inequality states that <math>{x^2 \ge 0}</math> for all x. This is a rather useful inequality for proving that certain quantities are non-negative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique. | ||
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==Applications== | ==Applications== | ||
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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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+ | == Intermediate == | ||
+ | === AIME 1992, Problem 13 === | ||
''Triangle <math>ABC</math> has <math>AB</math><math>=9</math> and <math>BC: AC=40: 41</math>. What's the largest area that this triangle can have?'' | ''Triangle <math>ABC</math> has <math>AB</math><math>=9</math> and <math>BC: AC=40: 41</math>. What's the largest area that this triangle can have?'' |
Revision as of 01:49, 18 June 2006
The Inequality
The trivial inequality states that for all x. This is a rather useful inequality for proving that certain quantities are non-negative. The inequality appears to be obvious and unimportant, but it can be a very powerful problem solving technique.
Applications
Maximizing and minimizing quadratic functions
After Completing the square, the trivial inequality can be applied to determine the extrema of a quadratic function.
Intermediate
AIME 1992, Problem 13
Triangle has
and
. What's 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 credit to: 4everwise