Let and , where . Let for all . Statement I: for all , because Statement II: is an…

MATHEMATICS · Subjective · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Let

u=\sqrt{c+1}-\sqrt{c}

and

v=\sqrt{c}-\sqrt{c-1}

, where

c>1

. Let

f(x)=\ln(1+x)

for all

x\in(-1,\infty)

. Statement I:

f(u)>f(v)

for all

c>1

, because Statement II:

f(x)

is an increasing function; hence, for

u>v

f(u)>f(v)