# More Proofs

• Nov 21st 2008, 06:40 AM
Caity
More Proofs
Let $\displaystyle \alpha > 0$. Prove that no matter how small $\displaystyle \alpha$ is, there is an $\displaystyle N \epsilon$ R such that $\displaystyle ln x \leq x^{\alpha}$ for all $\displaystyle x \leq N$. Note: N depends on $\displaystyle \alpha$.

I need to use the mean value theorem to do this.
I started by finding the difference of the derivatives of both functions.
Then didnt know where to go... Any help would be greatly appreciated...
• Nov 21st 2008, 08:55 AM
Caity
Fixed Latex error... sorry about that
• Nov 24th 2008, 10:04 AM
Caity
I've been playing around with this problem... This is what I have so far...

$\displaystyle g' (x) - f' (x)$ is decreasing when $\displaystyle \alpha$ is very small.

For $\displaystyle \alpha = \frac{1}{200}$ or smaller $\displaystyle x = 2.27$

The inequality only holds true for $\displaystyle x\leq 2.27$and I need to find a $\displaystyle N\leq x$

Can anyone help me with explaining this or let me know if I am doing something wrong please??