Question

The set of values of a for which the function $f\left(x\right)=(4a-3)(x+ln5)+2(a-7)cot\left(\frac{x}{2}\right)si{n}^2\left(\frac{x}{2}\right)$ does not possess critical point is

• A. $(1,\infty )$
• B. $[1,\infty )$
• C. $(-\infty ,2)$
• D. $(-\infty ,\frac{-4}{3})\cup (2,\infty )$

Answer 1

The correct option is D $(-\infty ,\frac{-4}{3})\cup (2,\infty )$

$\because f^{\prime }\left(x\right)=(4a-3)\left(1\right)+(a-7)cosx$
$=(\frac{4a-3}{(a-7)}+cosx)(a-7)$
$f^{\prime }\left(x\right)\ne 0$
(for non existence of critical points)
$\frac{4a-3}{a-7}>1or\frac{4a-3}{a-7}<-1(\because -1\le cosx\le 1)$
$\Rightarrow \frac{3a+4}{a-7}>0or\frac{5a-10}{a-7}<0$
$\therefore aϵ(-\infty ,-\frac{4}{3})\cup (7,\infty )oraϵ(2,7)$
$Hence,aϵ(-\infty ,-\frac{4}{3})\cup (2,\infty )(\because ata=7,f^{\prime }(x)\ne 0)$