The difference between the maximum and minimum values of the function `f(x)=sin^(3)x-3sinx, AA x in [0,(pi)/(6)]` is

Options

A. 2
B. `(1)/(2)`
C. `(11)/(8)`
D. `(7)/(6)`

Answer

To find the difference between the maximum and minimum values of the function \( f(x) = \sin^3 x - 3 \sin x \) for \( x \) in the interval \([0, \frac{\pi}{6}]\), we will follow these steps:

### Step 1: Substitute \( \sin x \) with \( t \)
Let \( t = \sin x \). Since \( x \) ranges from \( 0 \) to \( \frac{\pi}{6} \), we find the corresponding values of \( t \):
- When \( x = 0 \), \( t = \sin(0) = 0 \)
- When \( x = \frac{\pi}{6} \), \( t = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)

Thus, \( t \) ranges from \( 0 \) to \( \frac{1}{2} \).

### Step 2: Rewrite the function in terms of \( t \)
The function becomes:
\[ f(t) = t^3 - 3t \]

### Step 3: Find the derivative of the function
To find the critical points, we need to differentiate \( f(t) \):
\[ f'(t) = 3t^2 - 3 \]
Setting the derivative equal to zero to find critical points:
\[ 3t^2 - 3 = 0 \implies t^2 = 1 \implies t = \pm 1 \]
However, since \( t \) is restricted to the interval \([0, \frac{1}{2}]\), we only consider \( t = 1 \) which is outside our range.

### Step 4: Evaluate the function at the endpoints of the interval
Since there are no critical points in the interval, we evaluate \( f(t) \) at the endpoints:
- At \( t = 0 \):
\[ f(0) = 0^3 - 3 \cdot 0 = 0 \]
- At \( t = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 - 3 \cdot \left(\frac{1}{2}\right) = \frac{1}{8} - \frac{3}{2} = \frac{1}{8} - \frac{12}{8} = -\frac{11}{8} \]

### Step 5: Determine the maximum and minimum values
- Maximum value: \( f(0) = 0 \)
- Minimum value: \( f\left(\frac{1}{2}\right) = -\frac{11}{8} \)

### Step 6: Calculate the difference between maximum and minimum values
The difference between the maximum and minimum values is:
\[ \text{Difference} = \text{Maximum} - \text{Minimum} = 0 - \left(-\frac{11}{8}\right) = \frac{11}{8} \]

### Final Answer
The difference between the maximum and minimum values of the function \( f(x) \) in the interval \([0, \frac{\pi}{6}]\) is \( \frac{11}{8} \).

---