Difference between revisions of "Trivial Inequality"
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Revision as of 15:11, 26 October 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.
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
has 
and 
. What is the largest area that this triangle can have? 
, 
, and 
.
.
, the height, with 
being the base. Simplifying gives 
. To maximize 
. So if we can write: 
then 
is the maximum value for 
for 
. So we can keep increasing the left hand side of our earlier equation until 
. We can factor 
. We find 
. Thus, the area is 
.
