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Дьявол
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Homework Statement
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 ⊆ Df
Homework Equations
a) monotonically increasing if the set E ⊆ Df for arbitrary numbers x1, x2 ∊ E and x1<x2 ⇒ f(x1)<f(x2)
b)monotonically decreasing if the set E ⊆ Df for arbitrary numbers x1, x2 ∊ E and x1<x2 ⇒ f(x1)>f(x2)
The Attempt at a Solution
So we need to start with x1<x2. Now:
f(x1)-f(x2)=ln(x1-1)-ln(x2-1)=
=[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(x1)-f(x2)<0 and f(x1)<f(x2) and the function is monotonically increasing. Is this correct? Can I always use this method?
Thanks in advance.