The period of `sin""(pi[x])/12+cos""(pi[x])/4+tan""(pi[x])/3`, where [x] represents the greatest integer less than or equal to x is

Options

A. 12
B. 4
C. 3
D. 24

Answer

To find the period of the function \( f(x) = \sin\left(\frac{\pi [x]}{12}\right) + \cos\left(\frac{\pi [x]}{4}\right) + \tan\left(\frac{\pi [x]}{3}\right) \), where \([x]\) represents the greatest integer less than or equal to \(x\), we will analyze the periods of each individual trigonometric function involved.

### Step 1: Determine the period of \( \sin\left(\frac{\pi [x]}{12}\right) \)

The general form for the period of \( \sin(kx) \) is given by:

\[ \text{Period} = \frac{2\pi}{|k|} \]

In our case, \( k = \frac{\pi}{12} \), thus:

\[ \text{Period of } \sin\left(\frac{\pi [x]}{12}\right) = \frac{2\pi}{\left|\frac{\pi}{12}\right|} = \frac{2\pi}{\frac{\pi}{12}} = 2 \cdot 12 = 24 \]

### Step 2: Determine the period of \( \cos\left(\frac{\pi [x]}{4}\right) \)

Using the same formula for the cosine function, where \( k = \frac{\pi}{4} \):

\[ \text{Period of } \cos\left(\frac{\pi [x]}{4}\right) = \frac{2\pi}{\left|\frac{\pi}{4}\right|} = \frac{2\pi}{\frac{\pi}{4}} = 2 \cdot 4 = 8 \]

### Step 3: Determine the period of \( \tan\left(\frac{\pi [x]}{3}\right) \)

For the tangent function, the period is given by:

\[ \text{Period} = \frac{\pi}{|k|} \]

Here, \( k = \frac{\pi}{3} \):

\[ \text{Period of } \tan\left(\frac{\pi [x]}{3}\right) = \frac{\pi}{\left|\frac{\pi}{3}\right|} = \frac{\pi}{\frac{\pi}{3}} = 3 \]

### Step 4: Find the least common multiple (LCM) of the periods

Now we have the periods of the three functions:

- Period of \( \sin\left(\frac{\pi [x]}{12}\right) = 24 \)
- Period of \( \cos\left(\frac{\pi [x]}{4}\right) = 8 \)
- Period of \( \tan\left(\frac{\pi [x]}{3}\right) = 3 \)

To find the overall period of the function \( f(x) \), we need to calculate the LCM of these periods: 24, 8, and 3.

The prime factorization of each number is:

- \( 24 = 2^3 \times 3^1 \)
- \( 8 = 2^3 \)
- \( 3 = 3^1 \)

The LCM takes the highest power of each prime:

- For \( 2 \): highest power is \( 2^3 \)
- For \( 3 \): highest power is \( 3^1 \)

Thus, the LCM is:

\[ \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24 \]

### Conclusion

The period of the function \( f(x) = \sin\left(\frac{\pi [x]}{12}\right) + \cos\left(\frac{\pi [x]}{4}\right) + \tan\left(\frac{\pi [x]}{3}\right) \) is \( \boxed{24} \).

---