Prove f(x)=g(x)

Chris(DE)

Homework Statement

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]

Homework Equations

We are studying continuous functions and only have 3 theorems. IVT, Boundeness and the fact there is a max value for x.

The Attempt at a Solution

I am having trouble with this function for no other reason but I don't know how to state things.

First I drew out the problem and saw that f(x)=g(x) for some x.

This has to happen based on the IVT at some point the two graphs must cross. I also know that f(x)-g(x)= 0 My issue again is how to put this in a proof that would hold water.

I look at some value f(c) that is between f(a) and f(b). I can do this because of IVT. This would be a point that lies in [a,b] and would map to c. That same point should also be part of g since g is continous on the same interval. How can I argue that g(c)=f(c).

Sorry for sounding dumb. I am having trouble just putting it in words of a valid proof.

Thanks

what can you tell me of the function h(x)=g(x)-f(x) in [a,b]?

ZioX
You were nearly at your solution!

A lot of tricks dealing with MVTs and similar problems involve using an auxiliary equation to get what you want.