I am trying to solve an existence of fixed point problem.
I need to show that a function f (on reals) with f'(x)=>2 has a fixed
point.
I started off with g(x)=f(x)-x and the mean value theorem on g, and
need
to see if g(x)=0 has a solution.
I wish to show g(x)>0 for some x, and g(y)<0 for some y. And then
by
intermediate value theorem, there is a c such that g(c)=0 which implies
g(c)=f(c)-c=0 and hence f(c)=c. Thus, we have a fixed point.
But I am not able to show g(x)>0 and g(y)<0 for some x, y. Can you help?
Hi Bob,
g'(x) = f'(x) - 1 >= 1, so g(x) goes to - infinity when x goes to - infinity, (by the mean value theorem, g(-x) <= g(0) - x for x positive) and g(x) goes to infinity when x goes to infinity. This should help.
Claude