Question

The range of $a\in R$ for which the function
$f\left(x\right)=(4a-3)(x+{\log }_{e}5)+2(a-7)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right),$ $x\ne 2n\pi ,n\in N$ has critical points, is:

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

Answer 1

The correct option is A $[-\frac{4}{3},2]$

$f\left(x\right)=(4a-3)(x+\ln 5)+2(a-7)(\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}.{\sin }^2\frac{x}{2})f\left(x\right)=(4a-3)(x+\ln 5)+(a-7)\sin x\Rightarrow f^{\prime }\left(x\right)=(4a-3)+(a-7)\cos x=0\Rightarrow \cos x=-\frac{(4a-3)}{a-7}\Rightarrow -1\le -\frac{(4-3)}{a-7}<1(\because 1\le \cos x\le 1)\Rightarrow -1<\frac{4a-3}{a-7}\le 1\Rightarrow \frac{4a-3}{a-7}-1\le 0\text{and}\frac{4a-3}{a-7}+1>0\Rightarrow a\in [-\frac{4}{3},7)\text{and}a\in (-\infty ,2)\cup (7,\infty )\Rightarrow \frac{-4}{3}\le a<2$