Trivial Inequality

The Inequality

The trivial inequality states that $x^2 >= 0$ 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 $f(x)=ax^2+bx+c$ be a function of degree two, that is, a quadratic function, where $a\in\{1,-1\}$ . If ${a}=1$ , then ${f}$ has only a minimum; if $a=-1$ , then ${f}$ has only a maximum. We can 'complete the square' in this function as follows:

$ax^2+bx+c = a\left(x+\frac{b}{2}\right)^2-a\cdot \frac{b^2}{4}+c$

By the trivial inequality, the minimum/maximum is then easily determined [too lazy to explain].

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Trivial Inequality

The Inequality

Applications