F(x)=-f(-x), and F(x)=f(x+ 1) It is known that f (x) is a function with a period of 1, which is odd function in a complete period, and f(0)=0.

According to the properties of odd function, f(x) is monotonic in a single period, and f' (1) >: 0, which is judged as monotonic increasing function.

Periodic function f(5)=f(-5)=f(0)=0.

The derivative function of a periodic function is also a periodic function.

f '(-5)= f '( 1) =f '(0)>0;

F '(x)= f '(-x), and f' (x) is an even function (in a single period).

F ''(x) =-f ''(-x), f'' (x) is odd function (in a single period), and f ''(-5) = f ''(0) =0.

So f (5) = f'' (-5) < F'(-5), answer B.

For example, the periodic function f(x)=tan(x),-1/2.

f(0)=0，f '(0)= 1，f ''(0) =0