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Question

Show that f(x)=2x+cot1x+log(1+x2x) is increasing in R.
 


Solution

We have 
f(x)=2x+cot1x+log(1+x2x)f(x)=2+(11+x2)+1(1+x2x)(121+x2.2x1)=211+x2+1(1+x2x).(x1+x2)1+x2=211+x211+x2=2+2x11+x21+x2=1+2x21+x21+x2
For increasing function, f(x)0
1+2x21+x21+x201+2x21+x2(1+2x2)21+x21+4x4+4x21+x24x4+3x20x2(4x2+3)0
 Which is true for any real value of x.
Hence, f(x) is increasing in R.
 

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