The set of all values of '$a$' for which the function, $f(x)=(a^2-3a+2)(\cos^2{x/4} - \sin^2{x/4}) + (a-1)x + sin1$ does not posses critical points is:
I first differented it to find $f'(x)$, then I tried to find out the condition where it becomes zero so that the answer will be $\mathrm{R} -$ (the range). This is because for critical points, the function shouldn't be zero or not defined or non differentiable. It seems differentiable and could not have infinite slope.
But after differenting, I got: $(a^2 - 3a +2)(-\dfrac {\sin{x/2}}{2}) + (a-1)$
I don't get how to proceed with two variables $x$ and $a$.
Notice that if $a=1$, then $f$ is a constant function.
Now, we focus on $a \ne 1$,
$$(a^2-3a+2)\left(- \frac{\sin x/ 2}{2}\right)+(a-1)=0$$
$$(a-1)(a-2)\left(- \frac{\sin x/ 2}{2}\right)+(a-1)=0$$
$$(a-2) \left( - \frac{\sin x/ 2}{2}\right)+1=0$$
If $a=2$, then the equation has no solution.
Otherwise, $$\sin x/2 =- \frac{2}{a-2}$$
It doesn't have a solution if if $$\frac{2}{|a-2|} > 1$$
Hopefully you can carry on from here.