Question

Suppose f and g are continuous on [a,b] and that f(a) < g(a), but f(b) > g(b). Prove that f(x) = g(x) for some x in [a,b]. (If your proof isn't very short, it's not the right one.)

Answer 1

let h(x)=f(x)-g(x)
h(a)=f(a)-g(a)<0
h(b)=f(b)-g(b)>0
Since h is continuous and h(a)<0 and h(b)>0 then there is a number c in between (a,b) such that h(c)=0
h(c)=f(c)-g(c)=0
implies f(c)=g(c)

Answer 2

correct