Well, simple f'=1/(1+x^2) so f'(c)=1/(1+c^2)=(arctg(b)-arctg(a))/(b-a)

c^2=(b-a)/(arctg(b)-arctg(a))-1

c=sqrt(((b-a)/(arctg(b)-arctg(a))-1)

I need to show that (b-a)/(arctg(b)-arctg(a))-1>ab, and I know that the maximum is attained when it equals pi/2, not sure how to continue from here, any hints?