The range of the function `f(x)=3|sin x|-2|cos x|` is :

Options

A. `[-2 sqrt13]`
B. [-2,3]
C. `[3,sqrt13]`
D. `[-3,2]`

Answer

To find the range of the function \( f(x) = 3|\sin x| - 2|\cos x| \), we will analyze the behavior of the function based on the properties of the sine and cosine functions.

### Step 1: Understand the components of the function
The function consists of two parts: \( 3|\sin x| \) and \( -2|\cos x| \).

- The term \( 3|\sin x| \) will always be non-negative since the absolute value is always greater than or equal to zero. The maximum value of \( |\sin x| \) is 1, thus the maximum value of \( 3|\sin x| \) is \( 3 \times 1 = 3 \).
- The term \( -2|\cos x| \) will always be non-positive because it is multiplied by -2. The maximum value of \( |\cos x| \) is also 1, thus the maximum value of \( -2|\cos x| \) is \( -2 \times 1 = -2 \).

### Step 2: Find the minimum value of \( f(x) \)
To find the minimum value of \( f(x) \), we need to maximize the negative term \( -2|\cos x| \). This occurs when \( |\cos x| \) is at its maximum, which is 1.

- When \( |\cos x| = 1 \), \( |\sin x| \) will be 0 (since \( \sin^2 x + \cos^2 x = 1 \)). Therefore, at this point:
\[ f(x) = 3|\sin x| - 2|\cos x| = 3(0) - 2(1) = -2. \]

### Step 3: Find the maximum value of \( f(x) \)
To find the maximum value of \( f(x) \), we need to consider when both terms contribute positively.

- The maximum value of \( 3|\sin x| \) is 3 (when \( |\sin x| = 1 \)), and the minimum value of \( -2|\cos x| \) is 0 (when \( |\cos x| = 0 \)). This occurs when \( \sin x = \pm 1 \) and \( \cos x = 0 \).

Thus, at this point:
\[ f(x) = 3(1) - 2(0) = 3. \]

### Step 4: Conclusion on the range
From the analysis:
- The minimum value of \( f(x) \) is \(-2\).
- The maximum value of \( f(x) \) is \(3\).

Therefore, the range of the function \( f(x) = 3|\sin x| - 2|\cos x| \) is:
\[ [-2, 3]. \]

### Final Answer
The range of the function \( f(x) = 3|\sin x| - 2|\cos x| \) is \([-2, 3]\).