Question

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

Answer 1

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+2x−2x ( 2+x ) 2 = 1 ( 1+x ) − 4 ( 2+x ) 2 = ( 2+x ) 2 −4x−4 ( 2+x ) 2 ( 1+x )

Further simplify the above expression.

dy dx = 4+ x 2 +4x−4x−4 ( 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.