Helping students gain understanding and self-confidence in algebra.
smad wrote:Let f and g be two continuous functions on the closed interval [0,1] such that for every x in [0,1] : f(x)<g(x).
Prove that there exists some number m>0 such that for every x in [0,1] : f(x)+m<g(x).
smad wrote:Well if f and g are both continuous than therefore h=g-f is also continuous as a difference of 2 continuous functions and h>0 because f(x)<g(x). And the extreme value theorem states that if h is a continuous function on a closed interval [a,b] then there exists a c and a d such that h(c)>=h(x)>=h(d).