Question

The fundamental period of $\sin \frac{2x}{3}+\cos 4x+|\tan 3x|+sgn(x^2+4x+15)$ is $k\pi$, then the value of $k$ is

• A. $1$
• B. $2$
• C. $3$
• D. $\frac{1}{2}$

Answer 1

The correct option is C $3$

$f\left(x\right)=\sin \frac{2x}{3}+\cos 4x+|\tan 3x|+sgn(x^2+4x+15)$
The period of $\sin \frac{2x}{3}=3\pi$
of $\cos 4x=\frac{\pi }{2}$
of $|\tan 3x|=\frac{\pi }{3}$
While, $sgn(x^2+4x+15)=1$ (a constant function)
(Since, $x^2+4x+15$ is always positive)
$\therefore$ The fundamental period of $f\left(x\right)=3\pi =k\pi$
Hence, option C.