Trivial Inequality
The trivial inequality states that for all real numbers
. 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 1992/13) 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 credit to: 4everwise