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Question

Show that y=log(1+x)2x1+x,x>1, is an increasing function of x throughout its domain


Solution

Given, y=log(1+x)2x(2+x)
On differentiating, we get dydx=ddx[log(1+x)2x2+x]=11+x(2+x)ddx(2x)2xddx(2+x)(2+x)2=11+x4+2x2x(2+x)2=11+x4(2+x)2=(2x+x)24(1+x)(1+x)(2+x)2=4+x2+4x44x[1+x](2+x)2=x2(1+x)(2+x)2
When xϵ(1,), then x22+x2>0 and (1+x)>0
y>0 when x>1
Hence, y is an increasing function throughout (x>1) its domain.


Mathematics

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