Inequality: $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$

In summary, the conversation discusses an inequality being studied, which is $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$. It explains how the inequality is solved by rearranging terms and using algebraic techniques, and mentions that the variable x must be greater than 0 and not equal to 1. The conversation also discusses the use of natural logarithms in simplifying the inequality and the significance of the solution, which represents the range of values that satisfy the inequality and helps to understand the relationship between the two expressions.
  • #1
anemone
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Prove that $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$.
 
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Solution of other:

For $x>0$, we define

$f(x)=\dfrac{(x^3-1)(x+1)}{(x^3+x)}-3\ln x=\dfrac{x^4+x^3-x-1}{(x^3+x)}-3\ln x$

Differentiating $f$ w.r.t. $x$, we have

$\begin{align*}f'(x)&=\dfrac{(4x^3+3x^2-1)(x^3+x)-(x^4+x^3-x-1)(3x^2+1)}{(x^3+x)^2}-\dfrac{3}{x}\\&=\dfrac{x^6+3x^4+4x^3+3x^2+1}{(x^3+x)^2}-\dfrac{3}{x}\\&=\dfrac{x^6-3x^5+3x^4-2x^3+3x^2-3x+1}{(x^3+x)^2}---(*)\end{align*}$

The polynomial $P(x)$ in the numerator on the right in (*) has the property that $P(x)=x^6P\left(\dfrac{1}{x}\right)$ and so $x^{-3}P(x)$ can be written as a cubic polynomial in $x+\dfrac{1}{x}$.

We have

$f'(x)=\dfrac{x^3}{(x^3+x)^2}\left(\left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right)+4\right)$

as $x^3-3x+4=(x+1)(x-2)^2$, we obtain

$\begin{align*}f'(x)&=\dfrac{x^2+x+1}{(x^2+1)^2}\left(\left(x+\dfrac{1}{x}\right)-2\right)^2\\&=\dfrac{\left(\left(x+\dfrac{1}{x}\right)^2+\dfrac{3}{4}\right)(x-1)^2}{x^2(x^2+1)^2}\end{align*}$

so that $f'(x)>0$ for all $x>0$ while $f'(1)=0$.

Thus, $f(x)$ is a strictly increasing function of $x$ for all $x>0$. Hence in particular we have $f(x)>f(1)$ for $x>1$, and so

$\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}---(1)$ for $x>1$.

Replacing $x$ by $\dfrac{1}{x}$ in (1) we get

$\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}---(2)$ for $0<x<1$.

Inequalities (1) and (2) give the required result.
 

FAQ: Inequality: $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$

What is the inequality being studied?

The inequality being studied is $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$.

How is this inequality solved?

This inequality is solved by rearranging the terms and using algebraic techniques to isolate the variable on one side of the inequality sign.

What are the restrictions on the variable x?

The restrictions on the variable x are that it must be greater than 0 and not equal to 1.

How is the natural logarithm used in this inequality?

The natural logarithm is used to represent the inverse relationship between x and ln x, which helps to simplify the inequality and make it easier to solve.

What is the significance of the solution to this inequality?

The solution to this inequality represents the range of values that satisfy the inequality and can be used to understand the relationship between the two expressions on either side of the inequality sign.

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