(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]a>b>0[/tex] and let [tex]n \in \mathbb{N}[/tex] satisfy [tex]n \geq 2[/tex]. Prove that [tex]a^{1/n} - b^{1/n} < (a-b)^{1/n}[/tex].

[Hint: Show that [tex]f(x):= x^{1/n}-(x-1)^{1/n}[/tex] is decreasing for [tex]x\geq 1[/tex], and evaluate [tex]f[/tex] at 1 and a/b.]

2. Relevant equations

I assume, since this exercise is at the end of the Mean Value Theorem section, I am to use the Mean Value Theorem.

3. The attempt at a solution

I can show what the hint suggests. I guess I'm not sure how those ideas help exactly.

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# Homework Help: Mean Value Theorem exercise (Analysis)

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