Difference between revisions of "Trivial Inequality"

Revision as of 15:36, 22 April 2021

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 $x$ , $x^2 \ge 0$ , equality holds if and only if $x = 0$ .

Proof

We proceed by contradiction. Suppose there exists a real $x$ such that $x^2<0$ . We can have either $x=0$ , $x>0$ , or $x<0$ . If $x=0$ , then there is a clear contradiction, as $x^2 = 0^2 \not < 0$ . If $x>0$ , then $x^2 < 0$ gives $x < \frac{0}{x} = 0$ upon division by $x$ (which is positive), so this case also leads to a contradiction. Finally, if $x<0$ , then $x^2 < 0$ gives $x > \frac{0}{x} = 0$ upon division by $x$ (which is negative), and yet again we have a contradiction.

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.

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 $x^2+y^4\geq 2x+4y^2-4$ for all real $x$ .

Intermediate

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

Olympiad

Let $c$ be the length of the hypotenuse of a right triangle whose two other sides have lengths $a$ and $b$ . Prove that $a+b\le c\sqrt{2}$ . When does the equality hold? (1969 Canadian MO)

@@ Line 1: / Line 1: @@
-==The Inequality==
+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.
-The trivial inequality states that <math>{x^2 \ge 0}</math> 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.
+==Statement==
+For all [[real number]]s <math>x</math>, <math>x^2 \ge 0</math>, equality holds if and only if <math>x = 0</math>.
+==Proof==
+We proceed by contradiction. Suppose there exists a real <math>x</math> such that <math>x^2<0</math>. We can have either <math>x=0</math>, <math>x>0</math>, or <math>x<0</math>. If <math>x=0</math>, then there is a clear contradiction, as <math>x^2 = 0^2 \not < 0</math>. If <math>x>0</math>, then <math>x^2 < 0</math> gives <math>x < \frac{0}{x} = 0</math> upon division by <math>x</math> (which is positive), so this case also leads to a contradiction. Finally, if <math>x<0</math>, then <math>x^2 < 0</math> gives <math>x > \frac{0}{x} = 0</math> upon division by <math>x</math> (which is negative), and yet again we have a contradiction.
+Therefore, <math>x^2 \ge 0</math> for all real <math>x</math>, as claimed.
 ==Applications==
-'''Maximizing and minimizing quadratic functions'''
+The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
-After [[Completing the square]], the trivial inequality can be applied to determine the extrema of a quadratic function.
+One application is maximizing and minimizing [[quadratic]] functions. It gives an easy proof of the two-variable case of the [[AMGM | Arithmetic Mean-Geometric Mean]] inequality:
-==USA AIME 1992, Problem 13==
+Suppose that <math>x</math> and <math>y</math> are nonnegative reals. By the trivial inequality, we have <math>(x-y)^2 \geq 0</math>, or <math>x^2-2xy+y^2 \geq 0</math>. Adding <math>4xy</math> to both sides, we get <math>x^2+2xy+y^2 = (x+y)^2 \geq 4xy</math>. Since both sides of the inequality are nonnegative, it is equivalent to <math>x+y \ge 2\sqrt{xy}</math>, and thus we have <cmath> \frac{x+y}{2} \geq \sqrt{xy}, </cmath> as desired.
-''Triangle <math>ABC</math> has <math>AB=9</math> and <math>BC: AC=40: 41</math>. What's the largest area that this triangle can have?''
+== Problems ==
+===Introductory===
+*Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=4xy+6yz+2z-1</math>.
+*Show that <math>\sum_{k=1}^{n}a_k^2 \geq a_1a_2+a_2a_3+\cdots+a_{n-1}a_n+a_na_1</math>. [[Inequality_Introductory_Problem_2|Solution]]
+*Show that <math>x^2+y^4\geq 2x+4y^2-4</math> for all real <math> x </math>.
-Solution:
+===Intermediate===
-First, consider the triangle in a coordinate system with vertices at <math>(0,0)</math>, <math>(9,0)</math>, and <math>(a,b)</math>.<br>Applying the distance formula, we see that <math>\frac{ \sqrt{a^2 + b^2} }{ \sqrt{ (a-9)^2 + b^2 } } = \frac{40}{41}</math>.
+*Triangle <math>ABC</math> has <math>AB=9</math> and <math>BC: AC=40: 41</math>. What is the largest area that this triangle can have? ([[1992 AIME Problems/Problem 13|AIME 1992]])
-We want to maximize <math>b</math>, the height, with <math>9</math> being the base. Simplifying gives <math>-a^2 -\frac{3200}{9}a +1600 = b^2</math>. Recalling that the max/min value of a quadratic is <math>\frac{4ac-b^2}{4a}</math>, we see that <math>\frac{4(-1)(1600) - \left(\frac{3200}{9}\right)^2 }{(-1)(4)} = b^2</math>. Solving for <math>b</math>, <math>b = \frac{40\cdot41}{9}</math> Thus, the area is <math>9\cdot\frac{1}{2} \cdot \frac{40*41}{9} = 820</math>.
-''Solution credit to: 4everwise''
+===Olympiad===
+*Let <math>c</math> be the length of the [[hypotenuse]] of a [[right triangle]] whose two other sides have lengths <math>a</math> and <math>b</math>. Prove that <math>a+b\le c\sqrt{2}</math>. When does the equality hold? ([[1969 Canadian MO Problems/Problem 3|1969 Canadian MO]])
-Note: I am still editing this...
+[[Category:Inequality]]
+[[Category:Theorems]]

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