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Q.
For $a\,\in\,[\pi,2\pi]$ and $n\,\in\,Z$ ,the critical point of $f(x)=\frac{1}{3}\,\sin\,a \,\tan\,{}^3x+(\sin\,a-1)\,\tan\,x+\sqrt{\frac{a-2}{8-a}} $ are
Application of Derivatives
Solution:
$f'\left(x\right)=sin\,a\cdot tan^{2}\,x\,sec^{2}x+\left(sin\,a-1\right)sec^{2}x$
$=\left(sin\,a\cdot tan^{2}\,x+sin\,a-1\right)sec^{2}x$
At critical points, $f'\left(x\right)=0$
$\Rightarrow sin\,a\,tan^{2}\,x+sin\,a-1=0$
$[\because sec^2 x \ne 0$ for any $x \in R]$
$\Rightarrow tan^{2}\,x=\frac{1-sin\,a}{sin\,a}$
Since a $\in\left[\pi, 2\pi\right]$
$\therefore \frac{1-sin\,a}{sin\,a} < 0$
$\therefore $ the equation $tan^{2}\,x=\frac{1-sin\,a}{sin\,a}$ does not have a solution in $R$. Hence $f(x)$ has no critical points.