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

$\begin{array}{l}\text{Given function is }f\left(x\right)=\left(4a-3\right)\left(x+{\log }_{e}5\right)+2\left(a-7\right)\cot \left(\frac{x}{2}\right){\sin }^2\left(\frac{x}{2}\right),x\ne 2n\pi ,n\in N\\ \text{The above function is always continuous and differentiable in the domain}\\ \text{then }f\left(x\right)\text{ has criticial point if }{f}^{\text{'}}\left(x\right)=0\text{\hspace{0.17em}has soluton}\\ \text{Hence, }\left(4a-3\right)+\left(a-7\right)\cos x=0\\ \text{This equaiton has solution only when }cosx=\frac{4a-3}{7-a}\text{\hspace{0.17em}has solution}\\ \text{it implies that }-1\le \frac{3-4a}{a-7}\le 1\Rightarrow a\in \boxed{\left[-\frac{4}{3},2\right]}\end{array}$