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How do I continue to prove this?

Show that $$ x \ln(ex) - \sqrt{x}\geq 0 $$ for all $$ x \geq1 $$

My try:

$$\begin{eqnarray*} \\ \ln(e^x) + \ln(x^x) &\geq& \sqrt{x} \\ \\ \ln(e^x) &\geq& \sqrt{x} - \ln(x^x) \\ \\ e^x &\geq & e^\sqrt{x} e^{-\ln(x^x)} \\ \\ e^x&\geq& e^\sqrt{x} (1/x^x) \\ \\ e^x - \frac{e^\sqrt{x}}{x^x}&\geq& 0 \end{eqnarray*}$$

I "see" that this is bigger than 0, but I think that there are more calculations to do.

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    Let $x$ be positive but less than $1/e$. Then our expression is clearly negative. So you will have to change the condition $x\ge 0$ to something else.2012-09-27
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    @AndréNicolas He actually wants $x\geq 1$.2012-09-27
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    You have an error: $e^{-\ln x^x}\ne -x^x$; but rather $e^{-\ln x^x}= e^{\ln (1/x^x)}=1/x^x$.2012-09-27
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    @DavidMitra: Thanks.2012-09-27
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    Okay, but now I think it's obivous that this is bigger than 0?2012-09-27
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    You still need to correct the last two lines in your argument.2012-09-27
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    @DavidMitra: Yep, fixed that now.2012-09-27
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    Correction, meant x>=1!2012-09-27
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    @JulianAssange:I already mentioned that in my answer.2012-09-27
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    Then you might note $\ln(ex)\ge\ln(e)=1$; so, $x\ln(ex)-\sqrt x\ge x-\sqrt x=\sqrt x(\sqrt x-1)\ge 0$. Or, better, look at Peter's answer.2012-09-27

4 Answers 4

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Consider the function

$$f\left( x \right) = x\log \left( {ex} \right) - \sqrt x = x\log x + x - \sqrt x $$

$\log x$ is positive for $x>1$ and negative for $0.

And $x>\sqrt x$ for $x>1$, and $x<\sqrt x$ for $0. Thus

$$\begin{cases} f(x)>0 \text{ for } x>1\\ f(x)<0 \text{ for } 0

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    Sorry, but I meant x>=0.2012-09-27
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    @JulianAssange I'm showing you the inequality is not true for $x\geq 0$.2012-09-27
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    Sorry, I meant x >= 1. :)2012-09-27
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    @JulianAssange I showed you that is true, too. Did you read the answer carefully?2012-09-27
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    Yeah, I just wanted you to know. :=)2012-09-27
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Substitute 1/e for x. Of course 1/e is greater than zero. But ln(e*1/e) equals zero, so your Left Hand Side is negative, equals to -1/sqrt{e}. Therefore, the inequality you are trying to prove is false.

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    You should probably edit this to say that it was for the original version of the problem, which claimed the inequality for all $x\ge 0$.2012-09-27
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Let $f(x)=x\ln(ex)-\sqrt x$. Note that $f'(x)=\ln(ex)+1 -\tfrac{1}{2\sqrt x}>0 \;(*)$ for $x\geq1$, and so $f$ is increasing from $x=1$. But $f(1)=0$, and so $f(x)\geq 0$ for $x\geq 1$.

$(*)$ This can be seen since $\ln(ex)+1$ is increasing, and $\tfrac{1}{2\sqrt x}$ is decreasing, so $f'(x)$ must be increasing. Since $f'(1)>0$, we thus must have this inequality for all $x\geq1$.

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The inequality is true for $x \geq 1$. Put $f(x)=x\ln({\rm e}x)-\sqrt{x}$ and use the derivative test to prove that.

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    How does one prove that? I know that the function is acting like an exponential function, but is it enough to derivate and see that the interval from x >= 1 is growing?2012-09-27
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    @JulianAssange: Just look at any calculus book under applications of derivative (finding maximum and minimum of functions). I'll try to post it later.2012-09-27
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    @JulianAssange See my answer...2012-09-27