Given `A = {x : (pi)/(6) le x le (pi)/( 3)} and f(x) = cos x - x ( 1+ x )`. Find ` f (A)`.

Options

A. `[pi//6,pi//3]`
B. `[-pi//3,pi-6]`
C. `[(1)/(2)-(pi)/(3)(1+(pi)/(3)),(sqrt3)/(2)-(pi)/(6)(1+(pi)/(6))]`
D. `[(1)/(2)+(pi)/(3)(1-(pi)/(3)),(sqrt3)/(2)+(pi)/(6)(1-(pi)/(6))]`

Explanation

To find \( f(A) \) where \( A = \{ x : \frac{\pi}{6} \leq x \leq \frac{\pi}{3} \} \) and \( f(x) = \cos x - x(1 + x) \), we will follow these steps:

### Step 1: Rewrite the function
The function can be rewritten as:
\[ f(x) = \cos x - x - x^2 \]

### Step 2: Find the derivative
To determine the behavior of the function, we need to find its derivative:
\[ f'(x) = -\sin x - (1 + 2x) \]
This derivative will help us understand whether the function is increasing or decreasing in the interval.

### Step 3: Analyze the derivative
Since \( \sin x \) is positive in the interval \( \left[\frac{\pi}{6}, \frac{\pi}{3}\right] \), we can conclude that \( -\sin x \) is negative. Therefore, \( f'(x) \) is negative because:
\[ f'(x) = -\sin x - (1 + 2x) < 0 \]
This means that \( f(x) \) is a decreasing function in the interval \( \left[\frac{\pi}{6}, \frac{\pi}{3}\right] \).

### Step 4: Evaluate the function at the endpoints
Since the function is decreasing, we will find the maximum value at \( x = \frac{\pi}{6} \) and the minimum value at \( x = \frac{\pi}{3} \).

1. **Calculate \( f\left(\frac{\pi}{6}\right) \)**:
\[ f\left(\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) - \frac{\pi}{6}\left(1 + \frac{\pi}{6}\right) \]
\[ = \frac{\sqrt{3}}{2} - \frac{\pi}{6} - \frac{\pi^2}{36} \]

2. **Calculate \( f\left(\frac{\pi}{3}\right) \)**:
\[ f\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) - \frac{\pi}{3}\left(1 + \frac{\pi}{3}\right) \]
\[ = \frac{1}{2} - \frac{\pi}{3} - \frac{\pi^2}{9} \]

### Step 5: Determine the range
Since \( f(x) \) is decreasing, the maximum value of \( f(A) \) will be at \( f\left(\frac{\pi}{6}\right) \) and the minimum value will be at \( f\left(\frac{\pi}{3}\right) \).

Thus, the range of \( f(A) \) is:
\[ f(A) = \left[ f\left(\frac{\pi}{3}\right), f\left(\frac{\pi}{6}\right) \right] \]

### Final Result
The final result for \( f(A) \) is:
\[ f(A) = \left[ \frac{1}{2} - \frac{\pi}{3} - \frac{\pi^2}{9}, \frac{\sqrt{3}}{2} - \frac{\pi}{6} - \frac{\pi^2}{36} \right] \]