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