Trivial Inequality
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
Let be a function of degree two, that is, a quadratic function, where
. If
, then
has only a minimum; if
, then
has only a maximum. We can 'complete the square' in this function as follows:
By the trivial inequality, the minimum/maximum is then easily determined [too lazy to explain].