Difference between revisions of "Trivial Inequality"
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==The Inequality== | ==The Inequality== | ||
− | 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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''Solution credit to: 4everwise'' | ''Solution credit to: 4everwise'' | ||
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+ | == See also == | ||
+ | * [[inequalities]] | ||
+ | * [[optimization]] |
Revision as of 00:50, 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