if `f (x) sin {(pi)/(3) [X]-x^2}` for ` 2 lt x lt 3 ` and [X] denotes the greatest integer less than or equal to x then ` f' [sqrt(pi //3)] ` is equal to

Options

A. `- sqrt((pi)/(3))`
B. `sqrt(pi/3)`
C. `- sqrt(pi)/(2)`
D. `- sqrt( pi)`

Answer

To solve the problem, we need to find the derivative of the function \( f(x) = \sin\left(\frac{\pi}{3} [x] - x^2\right) \) for \( 2 < x < 3 \), and then evaluate \( f'(\sqrt{\frac{\pi}{3}}) \).

### Step 1: Determine the value of \([x]\)
Since \( 2 < x < 3 \), the greatest integer function \([x]\) will be equal to \(2\).

### Step 2: Substitute \([x]\) into the function
Now we substitute \([x] = 2\) into the function:
\[ f(x) = \sin\left(\frac{\pi}{3} \cdot 2 - x^2\right) = \sin\left(\frac{2\pi}{3} - x^2\right) \]

### Step 3: Differentiate the function
Now we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx} \left(\sin\left(\frac{2\pi}{3} - x^2\right)\right) \]
Using the chain rule:
\[ f'(x) = \cos\left(\frac{2\pi}{3} - x^2\right) \cdot \frac{d}{dx}\left(\frac{2\pi}{3} - x^2\right) \]
The derivative of \(\frac{2\pi}{3} - x^2\) is \(-2x\), so:
\[ f'(x) = -2x \cos\left(\frac{2\pi}{3} - x^2\right) \]

### Step 4: Evaluate \( f'(\sqrt{\frac{\pi}{3}}) \)
Now we substitute \( x = \sqrt{\frac{\pi}{3}} \) into the derivative:
\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cos\left(\frac{2\pi}{3} - \left(\sqrt{\frac{\pi}{3}}\right)^2\right) \]
Calculating \(\left(\sqrt{\frac{\pi}{3}}\right)^2\):
\[ \left(\sqrt{\frac{\pi}{3}}\right)^2 = \frac{\pi}{3} \]
Thus:
\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cos\left(\frac{2\pi}{3} - \frac{\pi}{3}\right) \]
This simplifies to:
\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cos\left(\frac{\pi}{3}\right) \]
We know that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\):
\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cdot \frac{1}{2} \]
This simplifies to:
\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -\sqrt{\frac{\pi}{3}} \]

### Final Answer
Thus, the value of \( f'(\sqrt{\frac{\pi}{3}}) \) is:
\[ \boxed{-\sqrt{\frac{\pi}{3}}} \]