Prove Existence: F(x) = (x-a)^2(x-b)^2 + x

[snipez90]

Homework Statement

$$Let F(x) = (x-a)^2(x-b)^2 + x$$. Show that the output $$\frac{a+b}{2}$$ exists for some value x.

Homework Equations

Quadratic formula. $$x^2 \geq 0$$.

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.