Proving Inequality with Positive Real Numbers $x,\,y,\,z$

In summary, proving inequality with positive real numbers involves using mathematical equations and proofs to show that one number is greater than another. This is important in science as it allows for comparisons and predictions to be made. Common methods include algebraic manipulations, induction, and contradiction, but it is important to avoid mistakes such as incorrect operations and assumptions without proper justification. This concept can be applied in various real-world situations, such as analyzing economic data and solving problems in science and engineering.
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anemone
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Let $x,\,y,\,z$ be positive real numbers such that $xy+yz+zx=3$.

Prove the inequality $(x^3-x+5)(y^5-y^3+5)(z^7-z^5+5)\ge 125$.
 
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Here is the solution that I found online and would like to share with MHB:

For any real numbers $a$, the numbers $a-1,\,a^2-1,\,a^3-1,\,a^5-1$ are of the same sign.

Therefore

$(a-1)(a^2-1)\ge 0$, $(a^2-1)(a^3-1)\ge 0$ and $(a^2-1)(a^5-1)\ge 0$.

i.e.

$x^3-x^2-x+1\ge 0$

$y^5-y^3-y^2+1\ge 0$

$z^7-z^5-z^2+1\ge 0$

So it follows that

$x^3-x+5\ge x^2+4$, $y^5-y^3+5\ge y^2+4$ and $z^7-z^5+5\ge z^2+4$

Multiplying these inequalities gives

$(x^3-x+5)(y^5-y^3+5)(z^7-z^5+5)\ge(x^2+4)(y^2+4)(z^2+4) \tag{1}$

We will prove that

$(x^2+4)(y^2+4)(z^2+4)\ge 25(xy+yz+xz+2) \tag{2}$

We have

$\begin{align*}(x^2+4)(y^2+4)(z^2+4)&=x^2y^2z^2+4(x^2y^2+y^2z^2+z^2x^2)+16(x^2+y^2+z^2)+64\\&=x^2y^2z^2+(x^2+y^2+z^2)+2+4(x^2y^2+y^2z^2+z^2x^2+3)+15(x^2+y^2+z^2)+50---(3)\end{align*}$

By the inequalities

$(x-y)^2+(y-z)^2+(z-x)^2\ge 0$ and

$(xy-1)^2+(yz-1)^2+(zx-1)^2\ge 0$ we obtain

$x^2+y^2+z^2\ge xy+yz+zx\tag{4}$ and

$x^2y^2+y^2z^2+z^2x^2+3\ge 2(xy+yz+zx)\tag{5}$

We will prove that

$x^2y^2z^2+(x^2+y^2+z^2)+2\ge 2(xy+yz+zx)\tag{6}$

Note that if we have $a,\,b,\,c>0$ then $3abc+a^3+b^3+c^3\ge 2((ab)^{\dfrac{3}{2}}+(bc)^{\dfrac{3}{2}}+(ca)^{\dfrac{3}{2}})$.

Its proof followed by Schur's inequality and AM-GM inequality.

For $a=x^{\dfrac{2}{3}}$, $b=y^{\dfrac{2}{3}}$ and $c=z^{\dfrac{2}{3}}$, we deduce

$3(xyz)^{\dfrac{2}{3}}+x^2+y^2+z^2\ge 2(xy+yz+zx)$

Therefore, it suffices to prove that

$x^2y^2z^2+2\ge 3(xyz)^{\dfrac{2}{3}}$, which follows immediately by $AM>GM$.

Thus we have proved inequality (6).

Now, by (3), (4), (5) and (6) we obtain inequality (2).

Finally by (1) and (2) and since $xy+yz+zx=3$ we obtain the required inequality.

Equality occurs iff $x=y=z=1$.
 

FAQ: Proving Inequality with Positive Real Numbers $x,\,y,\,z$

What is the definition of proving inequality with positive real numbers?

Proving inequality with positive real numbers involves showing that one number is greater than another number using positive real numbers as evidence. This is typically done using mathematical equations and proofs.

Why is proving inequality with positive real numbers important in science?

Proving inequality with positive real numbers is important in science because it allows us to make comparisons and draw conclusions about different quantities. It also helps us understand the relationship between different variables and make predictions based on this understanding.

What are some common methods used to prove inequality with positive real numbers?

Some common methods used to prove inequality with positive real numbers include using algebraic manipulations, induction, and contradiction. Other methods may also be used depending on the specific problem being addressed.

What are some common mistakes to avoid when proving inequality with positive real numbers?

Some common mistakes to avoid when proving inequality with positive real numbers include using incorrect mathematical operations, making assumptions without proper justification, and not providing enough evidence to support the inequality.

How can proving inequality with positive real numbers be applied in real-world situations?

Proving inequality with positive real numbers can be applied in real-world situations such as analyzing economic data, studying physical phenomena, and making decisions based on statistical data. It can also be used to solve problems in various fields of science and engineering.

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