Proving existence

[snipez90]

1. The problem statement, all variables and given/known data
Let F(x) = (x-a)^2(x-b)^2 + x. Show that the output \frac{a+b}{2} exists for some value x.

2. Relevant equations
Quadratic formula. x^2 \geq 0.


3. The attempt at a solution
Hmm I've tried setting the two equal but that doesn't look nice (if I multiply everything out). It's easy to find the zeros of F(x) so there might be someway to relate to that? If someone could just give me a hint at a good first step for showing the existence of a certain output of a function.

[snipez90]

Actually, I just edited it since there was an x in there. Now if a = b, then the output (a+b)/2 has to exist right? I'm not sure how to "show" it though. Show is just a bit more informal than a proof right?

[Dick]

F(a)=a and F(b)=b. That's a pretty good hint.

[snipez90]

So invoke the Intermediate Value Theorem?

[Dick]

So invoke the Intermediate Value Theorem?

Exactly.