Trivial Inequality

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The trivial inequality is an inequality that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness.

Statement

For all real numbers $x$, $x^2 \ge 0$.

Proof

We can have either $x=0$, $x>0$, or $x<0$. If $x=0$, then $x^2 = 0^2 \ge 0$. If $x>0$, then $x^2 = (x)(x) > 0$ by the closure of the set of positive numbers under multiplication. Finally, if $x<0$, then $x^2 = (-x)(-x) > 0,$ again by the closure of the set of positive numbers under multiplication.

Therefore, $x^2 \ge 0$ for all real $x$, as claimed.

Applications

The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.

One application is maximizing and minimizing quadratic functions. It gives an easy proof of the two-variable case of the Arithmetic Mean-Geometric Mean inequality:

Suppose that $x$ and $y$ are nonnegative reals. By the trivial inequality, we have $(x-y)^2 \geq 0$, or $x^2-2xy+y^2 \geq 0$. Adding $4xy$ to both sides, we get $x^2+2xy+y^2 = (x+y)^2 \geq 4xy$. Since both sides of the inequality are nonnegative, it is equivalent to $x+y \ge 2\sqrt{xy}$, and thus we have \[\frac{x+y}{2} \geq \sqrt{xy},\] as desired.

Another application will be to minimize/maximize quadratics. For example,

\[ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.\]

Then, we use trivial inequality to get $ax^2+bx+c\ge c-\frac{b^2}{4a}$ if $a$ is positive and $ax^2+bx+c\le c-\frac{b^2}{4a}$ if $a$ is negative.

Problems

Introductory

  • Find all integer solutions $x,y,z$ of the equation $x^2+5y^2+10z^2=4xy+6yz+2z-1$.
  • Show that $\sum_{k=1}^{n}a_k^2 \geq a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1$. Solution
  • Show that $x^2+y^4\geq 2x+4y^2-5$ for all real $x$ and $y$.

Intermediate

  • Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What is the largest area that this triangle can have? (AIME 1992)
  • The fraction,

\[\frac{ab+bc+ac}{(a+b+c)^2}\]

where $a,b$ and $c$ are side lengths of a triangle, lies in the interval $(p,q]$, where $p$ and $q$ are rational numbers. Then, $p+q$ can be expressed as $\frac{r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r+s$. (Solution here see problem 3 solution 1)

Olympiad

  • Let $x,y$ and $z$ be real numbers. Show that

\[(x^2+z^2)^2+y^4 \ge 4xzy^2\]

(Solution here see problem 13 solution 1)