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Monotone interval of economic mathematics
Find the stagnation point: y'=[2x( 1+x? )-2x? ]/( 1+x? )? =2x/( 1+x? )?

When y'=0 and x=0,

When x

So the function monotonically decreases at (negative infinity, 0) and monotonically increases at (0, positive infinity). The vertex is x=0.

y"=[2( 1+x? )? -8x? ( 1+x? )]/( 1+x? )^4=2( 1+x? )( 1-3x? )/( 1+x? )^4

1-3x? & gt0, that is, the interval on [-(root number 3)/3, (root number 3)/3] is concave, and other intervals are convex.

The inflection point is x=- (root number 3)/3, and x= (root number 3)/3.