Graphing Y=1/x + 1/(x+1): Problem with Curve Sketching Methods

[nameVoid]

in try to use curve sketching methods to graph this equation I've come across a problem in

y''=[2(x+1)^3+2x^3]/x^3(x+1)^3

I have verified this equation for y'' is correct but solving the numerator for critical numbers by hand ?

 

[MathematicalPhysicist]

How accurate does the graph need to be?

I mean you know how the graph 1/x looks like (I hope you do), 1/(x+1) is similar but with an asymptote in x=-1, the graph should be a superposition of these graphs, so if x+1>0, the graph of 1/(x+1) will tend more rapidly to zero than 1/x so it's dominant in the region of x>0, so the tendency of y to zero should be similar to 1/(x+1) from -1 to zero, and for 0<x<<1 the 1/x is dominant.

 

[Dick]

There several ways to solve x^3+(x+1)^3=0 by hand. i) Multiply it out and factor it. ii) There's a formula to factor the sum of two cubes. Or iii) write it as -x^3=(x+1)^3 and take the cube root. You'll lose the complex roots if you do it that way, but you don't need them for this problem anyway.