The range of a $in mathbb{R}$ for which the function
Question:

The range of $a \in \mathbb{R}$ for which the function

$f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$

$\mathrm{x}+2 \mathrm{n} \pi, \mathrm{n} \in \mathbb{N}$, has critical points, is

  1. (1) $(-3,1)$

  2. (2) $\left[-\frac{4}{3}, 2\right]$

  3. (3) $[1, \infty)$

  4. (4) $(-\infty,-1]$


Correct Option: , 2

Solution:

$f(x)=(4 a-3)\left(x+\log _{e} 5\right)+(a-7) \sin x$

$f(x)=(4 a-3)(1)+(a-7) \cos x=0$

$\cos x=\frac{3-4 a}{a-7}$

$\frac{3 a+4}{a-7} \leq 0$

$\frac{3-4 a}{a-7}+1 \geq 0$

$\frac{3-4 a+a-7}{a-7} \geq 0$

$\frac{-3 a-4}{a-7} \geq 0$

$\frac{3-4 a}{a-7}<1$

$\frac{3-4 a}{a-7}-1<0$

$\frac{3-4 a-a+7}{a-7}<0$

$\frac{5 a+10}{a-7}<0$

$\frac{5 a-10}{a-7}>0$

$\frac{5(a-2)}{a-7}>0$

$\alpha \in\left[-\frac{4}{3}, 2\right]$

Check end point $\left[-\frac{4}{3}, 2\right)$

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