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

Show that the function is monotonic, and if so find if it increases or decreases monotonically.

f(x)=ln(x-1), E=(1,∞) where E ⊆ D_{f}

2. Relevant equations

a) monotonically increasing if the set E ⊆ D_{f}for arbitrary numbers x_{1}, x_{2}∊ E and x_{1}<x_{2}⇒ f(x_{1})<f(x_{2})

b)monotonically decreasing if the set E ⊆ D_{f}for arbitrary numbers x_{1}, x_{2}∊ E and x_{1}<x_{2}⇒ f(x_{1})>f(x_{2})

3. The attempt at a solution

So we need to start with x_{1}<x_{2}. Now:

f(x_{1})-f(x_{2})=ln(x_{1}-1)-ln(x_{2}-1)=

=[tex]ln\frac{x_1-1}{x_2-1}=ln\frac{x_2-1+x_1-x_2}{x_2-1}=ln(1+\frac{x_1-x_2}{x_2-1})[/tex]

But I am stuck in here proving, so I tried:

[tex]x_1<x_2[/tex] ; [tex]x_1-1<x_2-1[/tex] ; [tex]\frac{x_1-1}{x_2-1}<\frac{x_2-1}{x_2-1}[/tex] ; [tex]\frac{x_1-1}{x_2-1}<1[/tex] ; [tex]ln\frac{x_1-1}{x_2-1}<ln(1)[/tex] ; [tex]ln\frac{x_1-1}{x_2-1}<0[/tex]

so f(x_{1})-f(x_{2})<0 and f(x_{1})<f(x_{2}) and the function is monotonically increasing. Is this correct? Can I always use this method?

Thanks in advance.

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# [prove] monotonicity of function

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