Question

The value of a for which the function $f\left(x\right)=(4a-3)(x+\log 5)+2(a-7)\cot \left(x/2\right){\sin }^2\left(x/2\right)$ does not possess critical points is

• A. $(-\infty ,-4/3)$
• B. $(-\infty ,-1)$
• C. $(1,\infty )$
• D. $(2,\infty )$

Answer 1

Answer: (A), (D)

The correct options are
A $(-\infty ,-4/3)$
D $(2,\infty )$

$f\left(x\right)=(4a-3)(x+\log 5)+2(a-7)\cot \left(x/2\right){\sin }^2\left(x/2\right)$
$\Rightarrow f\left(x\right)=(4a-3)(x+\log 5)+(a-7)\sin x$
$f\left(x\right)$ posses critical points when $f^{\prime }\left(x\right)=(4a-3)+(a-7)\cos x=0$
$\Rightarrow \cos x=-\frac{4a-3}{a-7}$
$\Rightarrow -1\le -\frac{4a-3}{a-7}\le 1$
$\Rightarrow -\frac{4}{3}\le a\le 2$
Therefore, $f\left(x\right)$ does not have critical points when $a\in (-\infty ,-4/3)\cup (2,\infty )$
Ans: A,D