What do you mean by "the sum"? Mathematics and vagueness do not mix very well.

Have you proven yet that:

$f(b) - f(a) = \dfrac{(a - b)(ab - 1)}{(1 + a^2)(1 + b^2)}$?

The point being, if we are to show it is positive, then both numerator and denominator have to have the same sign. Now the denominator is the product of two sums of squares. Squares are always positive, so the sum of squares is always positive, so the product in the denominator is positive. Therefore the whole fraction is positive if (and only if) the numerator is positive. How would you show that:

$a - b > 0$ and $ab - 1 > 0$?