Let fg be continuous function on ab Suppose fagb Prove that

Let f,g be continuous function on [a,b]. Suppose f(a)<g(a) and f(b)>g(b). Prove that there exists x sub zero belongs to (a,b) such that f(x sub zero)=g(x sub zero).

Solution

Define:

h(x)=f(x)-g(x)

h(a)=f(a)-g(a)<0

h(b)=f(b)-g(b)>0

h is continous on [a,b] because f and g are continous on [a,b]

Hence by INtermediate value theorem

h(x_0)=0 for some x_0 in (a,b)

ie f(x_0)-g(x_0)=0 for some x_0 in (a,b)