I'm looking at this youtube introduction on unconstrained optimization for one variable at this point in time and it says that if: f(x1) < f(x2) then we narrow the search range to [a, x2] instead of [a,b].
What if the top of the unimodal function f(x) lies between [x2,b]. Is that not still a possibility? So f(x1) < f(x2), but we have not reached the top of the summit yet?
She is describing the setting where $\exists \alpha \in [a,b]$ such that $f$ is strictly decreasing on $[a,\alpha]$ and strictly increasing on $[\alpha, b]$ and we are interested in minimization.
Hence, if $f(x_1) < f(x_2)$, then $x^*$ is in $[a,x_2]$ since the other values that are bigger than $x_2$ has a higher function value.