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Question

Show that , is an increasing function of x throughout its domain.

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Solution

The given function is y=log( 1+x ) 2x 2+x ,x>1.

Differentiate the function with respect to x.

dy dx = 1 ( 1+x ) ( 2+x )( 2 )2x ( 2+x ) 2 = 1 ( 1+x ) 4+2x2x ( 2+x ) 2 = 1 ( 1+x ) 4 ( 2+x ) 2 = ( 2+x ) 2 4x4 ( 2+x ) 2 ( 1+x )

Further simplify the above expression.

dy dx = 4+ x 2 +4x4x4 ( 2+x ) 2 ( 1+x ) = x 2 ( 2+x ) 2 ( 1+x )

Substitute the dy dx = 0.

x 2 ( 2+x ) 2 ( 1+x ) =0 x 2 =0 x=0 ( 2+x ) 2 0,( 1+x )0

Now, the point x=0 divides the domain ( 1, ) into two disjoint intervals given by,

1<x<0 x>0

When, 1<x<0,

x<0 x 2 >0 x>1( 2+x )>0 ( 2+x ) 2 >0 dy dx >0

When, x>0.

x>0 x 2 >0 ( 2+x ) 2 >0 dy dx >0

Thus, the function y=log( 1+x ) 2x 2+x ,x>1 is increasing throughout its domain.


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