`sqrt( 3) sin x + cos x ` is maximum when x is

Options

A. `30^(@)`
B. `90^(@)`
C. `60^(@)`
D. `45^(@)`

Answer

To find the value of \( x \) for which the expression \( \sqrt{3} \sin x + \cos x \) is maximum, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = \sqrt{3} \sin x + \cos x \).

### Step 2: Find the derivative
To find the maximum value, we need to differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \sqrt{3} \cos x - \sin x \]

### Step 3: Set the derivative to zero
To find the critical points, set the derivative equal to zero:
\[ \sqrt{3} \cos x - \sin x = 0 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ \sqrt{3} \cos x = \sin x \]

### Step 5: Divide both sides by \( \cos x \)
Assuming \( \cos x \neq 0 \), divide both sides by \( \cos x \):
\[ \sqrt{3} = \tan x \]

### Step 6: Solve for \( x \)
The equation \( \tan x = \sqrt{3} \) corresponds to specific angles. The principal angle where this occurs is:
\[ x = \frac{\pi}{3} + n\pi, \quad n \in \mathbb{Z} \]
However, for the maximum value in the range of \( [0, 2\pi] \), we take:
\[ x = \frac{\pi}{3} \]

### Conclusion
Thus, the value of \( x \) for which \( \sqrt{3} \sin x + \cos x \) is maximum is:
\[ x = \frac{\pi}{3} \text{ or } 60^\circ \]