The range of a $\in \mathbb{R}$ for which the function
$f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$
$\mathrm{x} \neq 2 \mathrm{n} \pi, \mathrm{n} \in \mathbb{N}$, has critical points, is :
Correct Option: , 2
$f(x)=(4 a-3)\left(x+\log _{e} 5\right)+(a-7) \sin x$
$f(x)=(4 a-3)(1)+(a-7) \cos x=0$
$\Rightarrow \quad \cos x=\frac{3-4 a}{a-7}$
$-1 \leq \frac{3-4 a}{a-7}<1$
$\frac{3-4 a}{a-7}+1 \geq 0 \quad \frac{3-4 a}{a-7}<1$
$\frac{3-4 a+a-7}{a-7} \geq 0 \quad \frac{3-4 a}{a-7}-1<0$
$\frac{-3 a-4}{a-7} \geq 0 \quad \frac{3-4 a-a+7}{a-7}<0$
$\frac{3 a+4}{a-7} \leq 0 \quad \frac{-5 a+10}{a-7}<0$
$\frac{5 a-10}{a-7}>0$
$\frac{5(a-2)}{a-7}>0$
$\alpha \in\left[-\frac{4}{3}, 2\right)$
Check end point $\left[-\frac{4}{3}, 2\right)$