Question

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

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

Answer 1

Answer: (D)

None of these

Given, $f\left(x\right)=\frac{1}{3}\sin a{\tan }^{3}x+(\sin a-1)\tan x+\sqrt{\frac{a-2}{8-a}}$

$f^{\prime }\left(x\right)=\sin a{\tan }^{2}x{\sec }^{2}x+(\sin a-1){\sec }^{2}x=(\sin a{\tan }^{2}x+\sin a-1){\sec }^{2}x$

At critical points, we must have $f^{\prime }\left(x\right)=0$

$\Rightarrow \sin a{\tan }^{2}x+\sin a-1=0(\because {\sec }^{2}x\ne 0foranyx\in R)$

$\Rightarrow {\tan }^{2}x=\frac{1-\sin a}{\sin a}$

Since $a\in [\pi ,2\pi ],\frac{1-\sin a}{\sin a}<0$

$\therefore {\tan }^{2}x=\frac{1-\sin a}{\sin a}$ has no solution in $R$

$\Rightarrow f\left(x\right)$ has no critical points