Trivial 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 1992/13) 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
