Mean Value Theorem exercise (Analysis)

[antiemptyv]

Homework Statement

Let a>b>0 and let n \in \mathbb{N} satisfy n \geq 2. Prove that a^{1/n} - b^{1/n} < (a-b)^{1/n}.
[Hint: Show that f(x):= x^{1/n}-(x-1)^{1/n} is decreasing for x\geq 1, and evaluate f at 1 and a/b.]

Homework Equations

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

The Attempt at a Solution

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

 

[morphism]

Did you evaluate f at 1 and a/b?

After you do that, keep in mind that a>b>0. You don't need to explicitly apply the mean value theorem (but you probably implicitly applied it when you proved that f(x) was decreasing for x>=1).

 

[antiemptyv]

Thanks for your quick reply. Let's see what we have here...

f(1)=1

and

f(\frac{a}{b}) = \frac{a^{1/n}-(a-b)^{1/n}}{b^{1/n}}.

 

[antiemptyv]

ohh, i think i see now.

1 < a/b

and, since f is decreasing,

f(\frac{a}{b}) = \frac{a^{1/n}-(a-b)^{1/n}}{b^{1/n}} < 1 = f(1)

and the rest is just algebra to show

a^{1/n} - b^{1/n} < (a-b)^{1/n}.

look good?