Practicing Success
For $a \in[\pi, 2 \pi]$ and $n \in Z$, the critical points of $f(x)=\frac{1}{3} \sin a \tan ^3 x+(\sin a-1) \tan x+\sqrt{\frac{a-2}{8-a}}$, are |
$x=n \pi$ $x=2 n \pi$ $x=(2 n+1) \pi$ none of these |
none of these |
We have, $f'(x)=\sin a+\tan ^2 x \sec ^2 x+(\sin a-1) \sec ^2 x$ $\Rightarrow f'(x)=\left(\sin a \tan ^2 x+\sin a-1\right) \sec ^2 x$ At critical points, we must have $f'(x)=0$ $\Rightarrow \sin a \tan ^2 x+\sin a-1=0$ [∵ sec2x ≠ 0 for any x ∈ R] $\Rightarrow \tan ^2 x=\frac{1-\sin a}{\sin a}$ Now, $a \in[\pi, 2 \pi]$ $\Rightarrow \frac{1-\sin a}{\sin a}<0 $ $\Rightarrow \tan ^2 x=\frac{1-\sin a}{\sin a}$ does not have solution in R. Hence, f(x) has no critical points. |