The range of $\mathrm{a}\in \mathrm{R}$ for which the function ${\text{⨍ (x)=(4a-3) (x\_log}}_{\mathrm{e}}\text{5)+2(a-7)cot}\left(\frac{\mathrm{x}}{2}\right){\text{sin}}^2\left(\frac{\mathrm{x}}{2}\right)\text{, x≠2nπ, n∈N}$ has critical points, is :

$$\begin{gathered} f\left(x\right)=(4a-3)(x-\log 5)+2(a-7)\frac{\cos {\displaystyle \frac{x}{2}}}{\sin {\displaystyle \frac{x}{2}}}{\sin }^2\frac{x}{2}=(4a-3)(x-\log 5)+(a-7)\sin x \\ f\text{'}\left(x\right)=4a-3+(a-7)\cos x \\ f\left(x\right) is min or max\Rightarrow f\text{'}\left(x\right)=0 \\ \cos x=\frac{-(4a-3)}{a-7} \\ f\left(x\right) has critcal points\Rightarrow \cos x is not defined \\ \cos x>1 \\ \frac{3-4a}{a-7}-1>0 \\ \frac{3-4a-a+7}{a-7}>0 \\ \frac{10-5a}{a-7}>0\Rightarrow \frac{a-2}{a-7}<0 \\ (a-2)(a-7)<0 \\ a\in (2,7) \\ \cos x>-1 \\ \frac{3-4a}{a-7}+1>0 \\ \frac{-4-3a}{a-7}>0 \\ \frac{3a+4}{a-7}<0 \\ (3a+4)(a-7)<0 \\ -\frac{4}{3}<a<7 \end{gathered}$$