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Difference between revisions of "Trivial Inequality"

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The '''Trivial Inequality''' states that <math>{x^2 \ge 0}</math> for all [[real number]]s <math>x</math>. 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.
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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.
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==Statement==
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For all [[real number]]s <math>x</math>, <math>x^2 \ge 0</math>.
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==Proof==
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We can have either <math>x=0</math>, <math>x>0</math>, or <math>x<0</math>. If <math>x=0</math>, then <math>x^2 = 0^2 \ge 0</math>. If <math>x>0</math>, then <math>x^2 = (x)(x) > 0</math> by the closure of the set of positive numbers under multiplication. Finally, if <math>x<0</math>, then <math>x^2 = (-x)(-x) > 0,</math> again by the closure of the set of positive numbers under multiplication.
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Therefore, <math>x^2 \ge 0</math> for all real <math>x</math>, as claimed.
  
 
==Applications==
 
==Applications==
  
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The trivial inequality can be used to maximize and minimize quadratic functions.
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The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
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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:
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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.
  
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After [[completing the square]], the trivial inequality can be applied to determine the extrema of a quadratic function.
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Another application will be to minimize/maximize quadratics. For example,
  
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Here is an example of the important use of this inequality:
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<cmath>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}.</cmath>
  
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Suppose that <math>a,b</math> are nonnegative [[real number]]s. Starting with <math>(a-b)^2\geq0</math>, after squaring we have <math>a^2-2ab+b^2\geq0</math>. Now add <math>4ab</math> to both sides of the inequality to get <math>a^2+2ab+b^2=(a+b)^2\geq4ab</math>. If we take the square root of both sides (since both sides are nonnegative) and divide by 2, we have the well-known [[AMGM | Arithmetic Mean-Geometric Mean]] Inequality for 2 variables: <math>\frac{a+b}2\geq\sqrt{ab}</math>
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Then, we use trivial inequality to get <math>ax^2+bx+c\ge c-\frac{b^2}{4a}</math> if <math>a</math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  
 
== Problems ==
 
== Problems ==
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=== Warm-up ===
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===Introductory===
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#Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=2z+6yz+4xy-1</math>.
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*Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=4xy+6yz+2z-1</math>.
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#*'''Solution:''' ''Challenge to the reader --[[User:10000th User|10000th User]] 12:58, 15 November 2007 (EST)''
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*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]]
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*Show that <math>x^2+y^4\geq 2x+4y^2-5</math> for all real <math>x</math> and <math>y</math>.
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===Intermediate===
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*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]])
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*The fraction,
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<cmath>\frac{ab+bc+ac}{(a+b+c)^2}</cmath>
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where <math>a,b</math> and <math>c</math> are side lengths of a triangle, lies in the interval <math>(p,q]</math>, where <math>p</math> and <math>q</math> are rational numbers. Then, <math>p+q</math> can be expressed as <math>\frac{r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r+s</math>. (Solution [[Problems Collection|here]] see problem 3 solution 1)
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===Olympiad===
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*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]])
  
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=== Intermediate ===
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*Let <math>x,y</math> and <math>z</math> be real numbers. Show that
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#Triangle <math>ABC</math> has <math>AB</math><math>=9</math> and <math>BC: AC=40: 41</math>. What is the largest area that this triangle can have? <div style='text-align:right;'>([[1992 AIME Problems/Problem 13|1992 AIME, Problem 13]])</div>
 
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#*'''Solution:''' 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>.
 
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#: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>. To maximize <math>b</math>, we want to maximize <math>b^2</math>. '''So if we can write: <math>-(a+n)^2+m=b^2</math> then <math>m</math> is the maximum value for <math>b^2</math>.''' This follows directly from the trivial inequality, because if <math>{x^2 \ge 0}</math> then plugging in <math>a+n</math> for <math>x</math> gives us <math>{(a+n)^2 \ge 0}</math>. So we can keep increasing the left hand side of our earlier equation until <math>{(a+n)^2 = 0}</math>. We can factor <math>-a^2 -\frac{3200}{9}a +1600 = b^2</math> into <math>-(a +\frac{1600}{9})^2 +1600+(\frac{3200}{9})^2 = b^2</math>. We find <math>b</math>, and plug into <math>9\cdot\frac{1}{2} \cdot b</math>. Thus, the area is <math>9\cdot\frac{1}{2} \cdot \frac{40*41}{9} = 820</math>.
 
  
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== See also ==
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<cmath>(x^2+z^2)^2+y^4 \ge 4xzy^2</cmath>
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* [[Optimization]]
 
  
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[[Category:Inequality]]
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(Solution [[Problems Collection|here]] see problem 13 solution 1)
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[[Category:Theorems]]
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[[Category:Algebra]]
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[[Category:Inequalities]]

Latest revision as of 20:20, 2 August 2024

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)

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