Let $f:[a,b]\rightarrow R$ be a continuous function. I have a general doubt regarding the continuous functions. I know that according to IMVT, this function will assume every value between $f(a)$ and $f(b)$. Now, Let, this function takes $f(a)=-1$, $f(\frac{a+b}{2})=1$ and $f(b)=-1$. Now, can we say(from these three values) that the range of this will be $[-1,1]$.
Thanks.
We can conclude that $[-1,1]$ is a subset of the range.
It is still possible for example to have $f\left( \frac{a+b}{4}\right) = 2.$ There is no restriction that $a$, $b$ and $\frac{a+b}2$ has to be the extreme values. We can construct such function simply by connecting those points with piecewise linear function.