The greatest value of the function `sqrt(3)sinx+cosx` is

Options

A. 4
B. 2
C. 1
D. `sqrt(3)`

Answer

To find the greatest value of the function \( f(x) = \sqrt{3} \sin x + \cos x \), we can follow these steps:

### Step 1: Rewrite the function
We can express the function in a different form to make it easier to analyze. We can factor out a constant:

\[ f(x) = \sqrt{3} \sin x + \cos x \]

### Step 2: Use the sine addition formula
We can rewrite \( f(x) \) using the sine addition formula. We know that:

\[ a \sin x + b \cos x = R \sin(x + \phi) \]

where \( R = \sqrt{a^2 + b^2} \) and \( \tan \phi = \frac{b}{a} \).

In our case, \( a = \sqrt{3} \) and \( b = 1 \).

### Step 3: Calculate \( R \)
First, we calculate \( R \):

\[ R = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 4: Calculate \( \phi \)
Next, we calculate \( \phi \):

\[ \tan \phi = \frac{1}{\sqrt{3}} \implies \phi = \frac{\pi}{6} \]

### Step 5: Rewrite the function
Now we can rewrite the function:

\[ f(x) = 2 \sin\left(x + \frac{\pi}{6}\right) \]

### Step 6: Find the maximum value
The maximum value of \( \sin \) function is 1. Therefore, the maximum value of \( f(x) \) is:

\[ f_{\text{max}} = 2 \cdot 1 = 2 \]

### Conclusion
Thus, the greatest value of the function \( \sqrt{3} \sin x + \cos x \) is:

\[ \boxed{2} \]