Trivial Inequality

The Trivial Inequality states that ${x^2 \ge 0}$ for all real numbers $x$ . This is a rather useful inequality for proving that certain quantities are nonnegative. 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.

Here is an example of the important use of this inequality:

Suppose that $a,b$ are nonnegative real numbers. Starting with $(a-b)^2\geq0$ , after squaring we have $a^2-2ab+b^2\geq0$ . Now add $4ab$ to both sides of the inequality to get $a^2+2ab+b^2=(a+b)^2\geq4ab$ . If we take the square root of both sides (since both sides are nonnegative) and divide by 2, we have the well-known Arithmetic Mean-Geometric Mean Inequality for 2 variables: $\frac{a+b}2\geq\sqrt{ab}$