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  4. If f(x)=(1/9)| cos x 1 0 1 2 cos x 1 0 1 2 cos x|, then (d2 f/d x2)=

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Q. If $f(x)=\frac{1}{9}\begin{vmatrix}\cos x & 1 & 0 \\ 1 & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x\end{vmatrix}$, then $\frac{d^{2} f}{d x^{2}}=$

AP EAMCETAP EAMCET 2019

Solution:

Given,
$f(x)=\frac{1}{9}\begin{vmatrix}\cos x & 1 & 0 \\ 1 & 2 \cos x & 1 \\ 0 & 1 & 2 \cos x\end{vmatrix}$,
$=\frac{1}{9}\left[\cos x\left(4 \cos ^{2} x-1\right)-1(2 \cos x)\right]$
$=\frac{1}{9}\left[4 \cos ^{3} x-\cos x-2 \cos x\right]$
$=\frac{1}{9}\left[4 \cos ^{3} x-3 \cos x\right]$
$f(x)=\frac{1}{9} \cos 3 x$
$\Rightarrow f^{'}(x)=\frac{1}{9}(-\sin 3 x) 3 $
$\Rightarrow f^{'}(x)=-\frac{1}{3} \sin 3 x$
$\Rightarrow f^{''}(x)=-\frac{1}{3}(\cos 3 x)(3)=-\cos 3 x$
$\Rightarrow f^{''}(x)=\cos (\pi+3 x) $
$\Rightarrow \frac{d^{2} f}{d v^{2}}=\cos (\pi+3 x)$