Periodicity of Functions

Solution Approach: We need to find the periods of $\sin(3x)$ and $\cos(2x)$ respectively, then find their least common multiple.

Detailed Steps:

1. Period of $\sin(3x)$: $T_1 = \frac{2\pi}{3}$
2. Period of $\cos(2x)$: $T_2 = \frac{2\pi}{2} = \pi$
3. Find the least common multiple: $\frac{2\pi}{3} = \frac{2\pi}{3}$, $\pi = \frac{3\pi}{3}$ Least common multiple is $2\pi$

Answer: The period of this function is $2\pi$.

Solution Approach: Simplify the function expression using trigonometric identities.

Detailed Steps:

1. Using the identity: $\sin^2 x + \cos^2 x = 1$
2. Therefore $f(x) = 1$, this is a constant function
3. A constant function is periodic, any non-zero real number is a period
4. The fundamental period does not exist (because any arbitrarily small positive number is a period)

Answer: This function is periodic but has no fundamental period.