If f(x) = | cases 2x & 2x & 2x - cos x & cos x & - sin x sin x &...

Expansion of determinant

$f\left(x\right) = \cos 2x + \cos 4x$

{f}^{'}( x ) = - 2\sin 2x - 4\text\sin4x = - 2\text\sin2x( 1 + 4\cos 2x )

$∴$ maxima at $x = 0$

{f}^{'}\left(x\right) = 0 ⇒ \text\sin 2 x = 0 ∴ x = 0 \text{,} \dfrac{\pi}{2} \text{,} - \dfrac{\pi}{2}
and $\cos 2x = - \dfrac{1}{4}$ has 4 solutions in \left(- \pi \text{,} \pi\right)$</p>$∴ {\text{f}}^{′} \left(x\right) = 0 has more than three solutions