Question

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

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

Answer 1

Answer: (D)

$1,2$

$f\left(x\right)=(a^2-2a-a+2)({\cos }^2\frac{x}{4}-{\sin }^2\frac{x}{4})$

$=(a-2)(a-1)\left(\cos \frac{x}{2}\right)f^{{}^{\prime }}\left(x\right)=(a-2)(a-1)(-sin\frac{x}{2})$

$f^{{}^{\prime }}\left(x\right)$ is $0$ when $a=2,1$.