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.