Let: g (x) = [2013 (x+1)+2012]/[2013 (x)+1]

We can consider the formal reason why the function g(x) is decomposed into the sum of a odd function and a constant: H (x) = x+sinx is odd function.

Settings:

G (x) = M(x)+t, where M(x) is odd function and t is a constant, then:

g(-x)=-M(x)+t

When the two formulas are added, you get:

2t=g(x)+g(-x)

First calculate:

g(x)+g(-x)=4025

De: 2t=4025

At this point, you will get:

F (x) = M(x)+t+h(x), where both M(x) and h(x) are odd function, then:

The maximum value m is obtained when x=k, and the minimum value n is obtained when x =-k.

Get:

M=f(k)=M(k)+t+h(k)、N=M(-k)+t+h(-k)=-M(k)+t-h(k)

Then:

M+N=2t=4025