Question

The set of all values of a for which the function $f(x)=(a^2-3a+2)({\cos }^{2}x/4-{\sin }^{2}x/4)+(a-1)x+\sin 1$ does not possess critical points is

• A. $[1.\infty )$
• B. $(0,1)\cup (1,4)$
• C. $(-2,4)$
• D. $(1,3)\cup (3,5)$

Answer 1

Answer: (B)

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