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As part of a proof in thermodynamics, I have arrived at the expression (which I use to define a function) $$f(x,N) := x^{1/N} + x^{-1/N} -2$$ where $x \geq 1$ and $N \in \mathbb{N}$. I want to show that this quantity is $\geq 0$ (and, on physical grounds associated with the total entropy change for this given cycle), I expect that if $x = 1$ and/or if $N \to \infty$, then $f(x,N) = 0$. However, I am struggling to manipulate things further. Can anyone either explain how to do it and/or provide a hint?

EE18
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1 Answers1

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Let $a=x^{\frac{1}{N}}$, then you have $a+\frac1a-2$

Using the Arithmetic-geometric mean: $$ a+\frac1a\geq 2\sqrt{a\cdot\frac1a}=2 $$ thus $$ x^{\frac{1}{N}}+x^{-\frac{1}{N}}-2\geq 0. $$ for all $x\geq 1$.

MZperX
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  • Thank you! Where does the $x\geq 1$ requirement enter here? – EE18 May 28 '23 at 19:27
  • That was the original condition in your post. – MZperX May 28 '23 at 19:31
  • I agree, I guess what I'm asking is where did you use it in your proof? – EE18 May 28 '23 at 20:04
  • This proof is valid for any positive real numbers, and it is used when we calculate the square root of a product, otherwise it wouldn't be defined. – Evil Witch May 28 '23 at 20:10
  • I see, so the only requirement actually used in this proof was $x\geq 0$? This makes sense since the process I was describing can be performed with final temperature less than initial temperature too $ 0 \leq x <1$. @Luchoming – EE18 May 28 '23 at 20:11
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    note that the inequality is strict, since when $x=0$ the function is not defined. – Evil Witch May 28 '23 at 20:13