Question

The set of values of a for which the function $f\left(x\right)=(4a-3)(x+In5)+2(a-7)cot\left(\frac{x}{2}\right)si{n}^2\frac{x}{2}$ 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)
$\frac{4a-3}{a-7}>1or\frac{4a-3}{(a-7)}<-1(\because -1\le cosx\le 1)\Rightarrow \frac{3a+4}{a-7}>0or\frac{5a-10}{a-7}<0\therefore aϵ(-\infty ,-\frac{4}{3})\cup (7,\infty )oraϵ(2,7)Hence,aϵ(-\infty ,-\frac{4}{3})\cup (2,\infty )(\because ata=7,f^{\prime }(x)\ne 0)$