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Q.
The set of all values of a for which the function $f(x) = (a^2 - 3a + 2) (\cos^2x/4 -\sin^2x/4) + (a -1) x + \sin 1$ does not possess critical points is
$f(x)=\left(a^{2}=-3 a+2\right)\left(\cos ^{2} x / 4-\sin ^{2} x / 4\right)+(a-1) \alpha+\sin 1$
$\Rightarrow f(x)=(a-1)(a-2) \cos \frac{x}{2}+(a-1) x+\sin 1$
$\Rightarrow f'(x)=\frac{-1}{2}(a-1)(a-2) \sin \frac{x}{2}+(a-1)$
$\Rightarrow f'(x)=(a-1)\left[1-\frac{(a-2)}{2} \sin \frac{x}{2}\right]$
If $f(x)$ does not possess critical points, then $f'(x) \neq 0$ for any $x \in R$.
$\Rightarrow (a-1)\left[1-\frac{(a-2)}{2} \sin \frac{x}{2}\right] \neq 0$ for any $x \in R$
$\Rightarrow a \neq 1$ and $1-\left(\frac{a-2}{2}\right) \sin \frac{x}{2}=0$
must not here any solution in $R$.
$\Rightarrow a \neq 1$ and $\sin \frac{x}{2}=\frac{2}{a-2}$ is not
solvable in $R$.
$\Rightarrow a \neq 1$ and $\left|\frac{2}{a-2}\right|>1$
[for $\left.a=2, f(x) \cdot x+\sin 1 \therefore f'(x)=1 \neq 0\right]$
$\Rightarrow a \neq 1$ and $|a-2| < 2$
$ \Rightarrow a \neq 1$ and $-2 < a-2 < 2$
$\Rightarrow a \neq 1$ and $0 < a < 4$
$ \Rightarrow a \in(0,1) \cup(1,4)$.