If `f(x)` and `g(x)` are periodic functions with the same fundamental period where `f(x)=sinalpha x+cos alpha x` and `g(x)=|sinx|+|cosx|`, then `alpha` is equal to (1) 0 (2) 2 (3) 4 (4) 8

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to find the value of `alpha` such that the functions `f(x)` and `g(x)` have the same fundamental period.

1. **Determine the period of `f(x)`**:
- The function `f(x) = sin(αx) + cos(αx)`.
- The period of `sin(αx)` is given by \( T_f = \frac{2\pi}{\alpha} \).
- The period of `cos(αx)` is also \( T_f = \frac{2\pi}{\alpha} \).
- Since both sine and cosine have the same period, the period of `f(x)` is:
\[ T_f = \frac{2\pi}{\alpha} \]

2. **Determine the period of `g(x)`**:
- The function `g(x) = |sin(x)| + |cos(x)|`.
- The period of `|sin(x)|` is \( \pi \) (since it repeats every π).
- The period of `|cos(x)|` is also \( \pi \).
- Therefore, the period of `g(x)` is:
\[ T_g = \pi \]

3. **Set the periods equal to find `alpha`**:
- Since both functions have the same fundamental period, we set their periods equal:
\[ \frac{2\pi}{\alpha} = \pi \]

4. **Solve for `alpha`**:
- To solve for `alpha`, we can cross-multiply:
\[ 2\pi = \alpha \cdot \pi \]
- Dividing both sides by π (assuming π ≠ 0):
\[ 2 = \alpha \]

Thus, the value of `alpha` is \( 2 \).

**Final Answer**: \( \alpha = 2 \) (Option 2)