Question

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

• A. $x=n\pi$
• B. $x=2n\pi$
• C. $x=(2n+1)\pi$
• D. no critical points

Answer 1

The correct option is D no critical points

$f\left(x\right)=\frac{1}{3}\sin a{\tan }^3\left(x\right)+(\sin a-1)\tan x+\sqrt{\frac{a-2}{8-a}}$
$f^{\prime }\left(x\right)={\sec }^{2}x(\sin a{\tan }^{2}x+\sin a-1)$
For critical points, $f^{\prime }\left(x\right)=0$
$\Rightarrow {\sec }^{2}x(\sin a{\tan }^{2}x+\sin a-1)=0$
$\Rightarrow {\tan }^{2}x=\frac{1-\sin a}{\sin a}$ [$\because {\sec }^{2}x=0$ (not possible)]
Since, $a\in [\pi ,2\pi ]$
$\Rightarrow \frac{1-\sin a}{\sin a}<0$
$\Rightarrow {\tan }^{2}x<0$ which is false.
Hence, no critical points.