Find the difference between the greatest and least values of the function `f(x) =sin 2x-x" on "[-(pi)/(2), (pi)/(2)]`.

Answer

To find the difference between the greatest and least values of the function \( f(x) = \sin(2x) - x \) on the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\), we will follow these steps:

### Step 1: Find the derivative of the function
We start by differentiating the function \( f(x) \):
\[ f'(x) = \frac{d}{dx}(\sin(2x) - x) = 2\cos(2x) - 1 \]

### Step 2: Set the derivative to zero to find critical points
Next, we set the derivative equal to zero to find the critical points:
\[ 2\cos(2x) - 1 = 0 \]
\[ \cos(2x) = \frac{1}{2} \]

### Step 3: Solve for \( x \)
The equation \( \cos(2x) = \frac{1}{2} \) gives us:
\[ 2x = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad 2x = -\frac{\pi}{3} + 2k\pi \quad (k \in \mathbb{Z}) \]
This simplifies to:
\[ x = \frac{\pi}{6} + k\pi \quad \text{or} \quad x = -\frac{\pi}{6} + k\pi \]

Considering the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\), we find the critical points:
- For \( k = 0 \): \( x = \frac{\pi}{6} \) and \( x = -\frac{\pi}{6} \)

### Step 4: Identify endpoints and critical points
We also need to evaluate the function at the endpoints of the interval:
- \( x = -\frac{\pi}{2} \)
- \( x = \frac{\pi}{2} \)

Thus, we have the points to evaluate:
1. \( x = -\frac{\pi}{2} \)
2. \( x = -\frac{\pi}{6} \)
3. \( x = \frac{\pi}{6} \)
4. \( x = \frac{\pi}{2} \)

### Step 5: Evaluate the function at these points
Now we evaluate \( f(x) \) at these points:

1. **At \( x = -\frac{\pi}{2} \)**:
\[ f\left(-\frac{\pi}{2}\right) = \sin(-\pi) + \frac{\pi}{2} = 0 + \frac{\pi}{2} = \frac{\pi}{2} \]

2. **At \( x = -\frac{\pi}{6} \)**:
\[ f\left(-\frac{\pi}{6}\right) = \sin\left(-\frac{\pi}{3}\right) + \frac{\pi}{6} = -\frac{\sqrt{3}}{2} + \frac{\pi}{6} \]
(Approximate value: \( -0.866 + 0.524 \approx -0.342 \))

3. **At \( x = \frac{\pi}{6} \)**:
\[ f\left(\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{3}\right) - \frac{\pi}{6} = \frac{\sqrt{3}}{2} - \frac{\pi}{6} \]
(Approximate value: \( 0.866 - 0.524 \approx 0.342 \))

4. **At \( x = \frac{\pi}{2} \)**:
\[ f\left(\frac{\pi}{2}\right) = \sin(\pi) - \frac{\pi}{2} = 0 - \frac{\pi}{2} = -\frac{\pi}{2} \]

### Step 6: Determine the greatest and least values
Now we compare the values:
- \( f\left(-\frac{\pi}{2}\right) = \frac{\pi}{2} \)
- \( f\left(-\frac{\pi}{6}\right) \approx -0.342 \)
- \( f\left(\frac{\pi}{6}\right) \approx 0.342 \)
- \( f\left(\frac{\pi}{2}\right) = -\frac{\pi}{2} \)

The greatest value is \( \frac{\pi}{2} \) and the least value is \( -\frac{\pi}{2} \).

### Step 7: Calculate the difference
The difference between the greatest and least values is:
\[ \text{Difference} = \frac{\pi}{2} - \left(-\frac{\pi}{2}\right) = \frac{\pi}{2} + \frac{\pi}{2} = \pi \]

### Final Answer
Thus, the difference between the greatest and least values of the function is:
\[ \boxed{\pi} \]