Critical points in a function

Question

Consider the function $f(x)=(a^2 -3a+2)\cos(x/2) + (a-1)x$. We have to find set of values of $a$ for which $f(x)$ possess critical points. When we put $a=1$, we get both $f(x) = 0$ and $f'(x) =0$. So we can say $f(x)$ possess critical points for $a=1$. But in the answer, $a=1$ is not included. Why?

Answer 1

Hints:

1. Compute $f'(x)$ and distinguish three cases: $a=1,a=2$ and $ 1 \ne a \ne 2$

2. Compute all $x$ with $f'(x)=0$