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?