If `A={x:(pi)/(6) lt x lt (pi)/(3)}` and `f(x) =cosx-x(1+x),` then `f(A)` is equal to :

Options

A. `[(pi),(6),(pi)/(3)]`
B. `[(-pi)/(3),(-pi)/(6)]`
C. `[(1)/(2)-(pi)/(3)(1+(pi)/(3)),(sqrt(3))/(2)-(pi)/(6)(1+(pi)/(6))]`
D. None of these

Answer

To solve the problem step by step, we need to find the range of the function \( f(x) = \cos x - x(1+x) \) over the interval \( A = \left\{ x : \frac{\pi}{6} < x < \frac{\pi}{3} \right\} \).

### Step 1: Define the function
The function is given as:
\[ f(x) = \cos x - x(1+x) \]
We can rewrite it as:
\[ f(x) = \cos x - x - x^2 \]

### Step 2: Find the derivative
To determine whether the function is increasing or decreasing, we need to find the derivative \( f'(x) \):
\[ f'(x) = -\sin x - (1 + 2x) \]
This simplifies to:
\[ f'(x) = -\sin x - 1 - 2x \]

### Step 3: Analyze the sign of the derivative
We need to check the sign of \( f'(x) \) in the interval \( \left( \frac{\pi}{6}, \frac{\pi}{3} \right) \).

1. Since \( \sin x \) is positive in this interval, \( -\sin x \) is negative.
2. The term \( -1 - 2x \) is also negative because \( 2x \) is positive for \( x > 0 \).

Thus, \( f'(x) < 0 \) in the interval \( \left( \frac{\pi}{6}, \frac{\pi}{3} \right) \). This means that \( f(x) \) is a decreasing function in this interval.

### Step 4: Calculate the function values at the endpoints
To find the range of \( f(x) \), we need to evaluate \( f(x) \) at the endpoints of the interval:

1. **At \( x = \frac{\pi}{6} \)**:
\[ f\left(\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) - \frac{\pi}{6}\left(1 + \frac{\pi}{6}\right) \]
We know that \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \), so:
\[ f\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} - \frac{\pi}{6}\left(1 + \frac{\pi}{6}\right) \]

2. **At \( x = \frac{\pi}{3} \)**:
\[ f\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) - \frac{\pi}{3}\left(1 + \frac{\pi}{3}\right) \]
We know that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), so:
\[ f\left(\frac{\pi}{3}\right) = \frac{1}{2} - \frac{\pi}{3}\left(1 + \frac{\pi}{3}\right) \]

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

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:
\[ f(A) = \left[ \frac{1}{2} - \frac{\pi}{3}\left(1 + \frac{\pi}{3}\right), \frac{\sqrt{3}}{2} - \frac{\pi}{6}\left(1 + \frac{\pi}{6}\right) \right] \]