Question

For $a\in [\pi ,2\pi ]$ and $n\in Z$ ,the critical point of $f(x)=\frac{1}{3}\sin a\tan {}^{3}x+(\sin a-1)\tan x+\sqrt{\frac{a-2}{8-a}}$ are

• A. $x=n\pi$
• B. $x=2n\pi$
• C. $x=(2n+1)\pi$
• D. none of these.

Answer 1

Answer: (D)

$f^{\prime }\left(x\right)=sina\cdot ta{n}^{2}xse{c}^{2}x+\left(sina-1\right)se{c}^{2}x$
$=\left(sina\cdot ta{n}^{2}x+sina-1\right)se{c}^{2}x$
At critical points, $f^{\prime }\left(x\right)=0$
$\Rightarrow sinata{n}^{2}x+sina-1=0$
$[\because se{c}^{2}x\ne 0$ for any $x\in R]$
$\Rightarrow ta{n}^{2}x=\frac{1-sina}{sina}$
Since a $\in \left[\pi ,2\pi \right]$
$\therefore \frac{1-sina}{sina}<0$
$\therefore$ the equation $ta{n}^{2}x=\frac{1-sina}{sina}$ does not have a solution in $R$. Hence $f(x)$ has no critical points.