Showing value exists with intermediate-value theorem

Question

I have this problem:

"Show that the function $f(x) = (x-a)^{2}(x-b) + x$ has a value $f(c) = \frac{a+b}{2}$ for a number c"

I am new to this kind of problems and I am having a bit of trouble expressing my answer. Can anyone give me some advice?

I want to write something like this:

"$f(x)$ is a polynomial and thus continuous everywhere,

$f(a) = a$,

$f(b) = b$

$a < \frac{a+b}{2} < b$

Thus, according to the intermediate-value theorem, $f(x)$ has a value $f(c)$ for a number $c$."

Can I write that? Sorry if it's obvious but I am not used to doing math this "proofy"- kind of way, and I find it hard to formulate it as a proper "question"

Answer 1

You are right to be a bit concerned - you have no reason to say that $a<b$!
However, it is true that, unless $a=b$, either $a<\frac{a+b}{2}<b$ or $b<\frac{a+b}{2}<a$.
This is because for any $a,b \in \mathbb{R}$ we have to have exactly one of $a<b$, $a=b$ or $b<a$ (this is an axiom), and in each case we can apply the theorem.