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
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