Application Of Derivatives Ques 1
- If $f(x)=\left|\begin{array}{ccc}\cos (2 x) & \cos (2 x) & \sin (2 x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x\end{array}\right|$, then
(2017 Adv.)
(a) $f(x)$ attains its minimum at $x=0$
(b) $f(x)$ attains its maximum at $x=0$
(c) $f^{\prime}(x)=0$ at more than three points in $(-\pi, \pi)$
(d) $f^{\prime}(x)=0$ at exactly three points in $(-\pi, \pi)$
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Answer:
Correct Answer: 1.(b,c)
Solution: 30. (b,c)
$ f(x)=$ $\left|\begin{array}{ccc} \cos 2 x & \cos 2 x & \sin 2 x \ -\cos x & \cos x & -\sin x \ \sin x & \sin x & \cos x \end{array}\right| $
$\cos 2 x\left(\cos ^2 x+\sin ^2 x\right)-\cos 2 x $ $\left(-\cos ^2 x+\sin ^2 x\right)+\sin 2 x(-\sin 2 x) $
$=\cos 2 x+\cos 4 x $
$f^{\prime}(x)=-2 \sin 2 x-4 \sin 4 x=-2 \sin 2 x(1+4 \cos 2 x) $
$\text { At } x=0 $
$f^{\prime}(x)=0 $
$\text { and } f(x)=2 $
$\text { Also, } f^{\prime}(x)=0 $
$\sin 2 x=0 $
$\text { or } \cos 2 x=\frac{-1}{4} $
$\Rightarrow x=\frac{n \pi}{2} \text { or } \cos 2 x=-\frac{1}{4}$
BETA
