proοf οf existence of roots

[George Saliaris]

Given f: [a,b] -> R a continuous function such that: a<f(x)<b for every x Ε [a,b].Let also h(x) a continuous function defined at [a,b] with h([a,b]) = [a, b]. Prove that equation f(x) + f(a+b-x) = 2h(x) has at least 1 solution at (a,b)..What I have tried so far :
1) Bolzano's theorem for w(x) =f(x) + f(a+b-x) - 2h(x) and I have stuck on proving that w(a) * w(b) < 0..Any suggestions?

 

[Dr.Peterson]

Given f: [a,b] -> R a continuous function such that: a<f(x)<b for every x Ε [a,b].Let also h(x) a continuous function defined at [a,b] with h([a,b]) = [a, b]. Prove that equation f(x) + f(a+b-x) = 2h(x) has at least 1 solution at (a,b)..What I have tried so far :
1) Bolzano's theorem for w(x) =f(x) + f(a+b-x) - 2h(x) and I have stuck on proving that w(a) * w(b) < 0..Any suggestions?

We don't know the values of w(a) and w(b); we are told nothing about what happens at the endpoints.

But we do know that there are some points in [a,b], say c and d, such that h(c) = a and h(d) = b. Can you use that?

 

[George Saliaris]

We don't know the values of w(a) and w(b); we are told nothing about what happens at the endpoints.

But we do know that there are some points in [a,b], say c and d, such that h(c) = a and h(d) = b. Can you use that?

You mean Intermediate value Theorem for h(x), right?

 

[Dr.Peterson]

You mean Intermediate value Theorem for h(x), right?

I might. There may be other ways, too, so I don't want to influence you too much.

 

[George Saliaris]

Do you mind being more elaborate,because I cannot think of any effective way of using what you mentioned previously?

 

[Dr.Peterson]

Do you mind being more elaborate,because I cannot think of any effective way of using what you mentioned previously?

Restrict yourself to the interval I described as [c,d] and do just what you originally planned, when you said, "I have stuck on proving that w(a) * w(b) < 0". Now it will be w(c)*w(d), which you can do.

 

[George Saliaris]

Oh yes, thanks for helping me ^_^.