Question

Let $f(x)=\left|\begin{array}{ccc}\sin 3x& 1& 2{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ \cos 3x& -1& 2\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ \tan 3x& 4& 1+2\tan 3x\end{array}\right|$
Then, the value of $f^{\prime }(x)$ at $x=(2n+1)\pi ,n\in I$ (the set of integers) is equal to

• A. $(-1{)}^n$
• B. $(-1{)}^{n+1}$
• C. $3$
• D. $9$

Answer 1

Answer: (C)

Given, $f(x)=\left|\begin{array}{ccc}<br/>\sin 3x& 1& 2{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ <br/>\cos 3x& -1& 2\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ <br/>\tan 3x& 4& 1+2\tan 3x<br/>\end{array}\right|$
On differentiating w.r.t. $x$, we get
$f^{\prime }(x)=\left|\begin{array}{ccc}\frac{d}{dx}(\sin 3x)& 1& 2{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ \frac{d}{dx}(\cos 3x)& -1& 2\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ \frac{d}{dx}(\tan 3x)& 4& 1+2\tan 3x\end{array}\right|$
$+\left|\begin{array}{ccc}\sin 3x& \frac{d}{dx}(1)& 2{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ \cos 3x& \frac{d}{dx}(-1)& 2\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ \tan 3x& \frac{d}{dx}(4)& 1+2\tan 3x\end{array}\right|$
$+\left|\begin{array}{ccc}\sin 3x& 1& 2\frac{d}{dx}{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ \cos 3x& -1& 2\frac{d}{dx}\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ \tan 3x& 4& \frac{d}{dx}(1+2\tan 3x)\end{array}\right|$
$=\left|\begin{array}{ccc}3\cos 3x& 1& 2{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ 3\sin 3x& -1& 2\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ 3{\sec }^{2}3x& 4& 1+2\tan 3x\end{array}\right|$
$+\left|\begin{array}{ccc}\sin 3x& 0& 2{\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)}^2\\ \cos 3x& 0& 2\left({\cos }^2\frac{3x}{2}-{\sin }^2\frac{3x}{2}\right)\\ \tan 3x& 0& 1+2\tan 3x\end{array}\right|$
$+\left|\begin{array}{ccc}\sin 3x& 1& 2\times 2\left(\cos \frac{3x}{2}+\sin \frac{3x}{2}\right)\times \left(\frac{-3}{2}\sin \frac{3x}{2}+\frac{3}{2}\cos \frac{3x}{2}\right)\\ \cos 3x& -1& 2\left(-2\cos \frac{3x}{2}\times \frac{3}{2}\sin \frac{3x}{2}-2\sin \frac{3x}{2}\times \frac{3}{2}\cos \frac{3x}{2}\right)\\ \tan 3x& 4& \left(0+2\times 3{\sec }^{2}3x\right)\end{array}\right|$
At $x=(2n+1)\pi$
$f^{\prime }(x)=\left|\begin{array}{ccc}3(-1)& 1& 2(1)\\ 0& -1& 2(-1)\\ 3& 4& 1+0\end{array}\right|+0$
$+\left|\begin{array}{ccc}0& 1& 4(0-1)\times \left[-\frac{3}{2}(-1)+\frac{3}{2}\times 0\right]\\ -1& -1& 0\\ 0& 4& 0\end{array}\right|$
$=[-3(-1+8)-1(0+6)+2(0+3)]+[0-1(0-0)-6(-4)]$
$=-21+24=3$