The function `f(x) = sin^3 x-m sin x` is defined on open interval `(-pi/2,pi/2)` and if assumes only 1 maximum value and only 1 minimum value on this interval. Then, which one of the must be correct? (a) `0lt m lt 3` (b) `-3 lt m lt 0` (c) `m gt 3` (d) `m lt -3`

Options

A. `0lt m lt 3`
B. `-3 lt m lt 0`
C. `m gt 3`
D. `m lt -3`

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin^3 x - m \sin x \) and determine the conditions under which it has only one maximum and one minimum value in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \):

\[ f'(x) = \frac{d}{dx}(\sin^3 x - m \sin x) \]

Using the chain rule and product rule, we get:

\[ f'(x) = 3 \sin^2 x \cos x - m \cos x \]

### Step 2: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:

\[ f'(x) = 0 \implies 3 \sin^2 x \cos x - m \cos x = 0 \]

Factoring out \( \cos x \):

\[ \cos x (3 \sin^2 x - m) = 0 \]

This gives us two cases:

1. \( \cos x = 0 \)
2. \( 3 \sin^2 x - m = 0 \)

### Step 3: Analyze the first case
The first case, \( \cos x = 0 \), occurs at \( x = \pm \frac{\pi}{2} \). However, these points are not included in the open interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). Therefore, we discard this case.

### Step 4: Analyze the second case
For the second case, we solve:

\[ 3 \sin^2 x - m = 0 \implies \sin^2 x = \frac{m}{3} \]

### Step 5: Determine the range of \( \sin^2 x \)
Since \( x \) is in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), the range of \( \sin x \) is \( (-1, 1) \). Therefore, \( \sin^2 x \) will be in the range \( (0, 1) \).

This means:

\[ 0 < \sin^2 x < 1 \]

Substituting \( \sin^2 x = \frac{m}{3} \):

\[ 0 < \frac{m}{3} < 1 \]

### Step 6: Solve the inequalities
Multiplying through by 3 gives:

\[ 0 < m < 3 \]

### Conclusion
Thus, the value of \( m \) must satisfy \( 0 < m < 3 \). The correct option that corresponds to this condition is:

**(a) \( 0 < m < 3 \)**