If `f(x)=sin{(pi)/(3)[x]-x^(2)}" for "2ltxlt3` 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)`
D. none of these

Explanation

To solve the problem, we need to find the value of \( f''\left(\sqrt{\frac{\pi}{3}}\right) \) for the function defined as:

\[ f(x) = \sin\left(\frac{\pi}{3}[x] - x^2\right) \]

where \([x]\) denotes the greatest integer less than or equal to \(x\), and \(2 < x < 3\).

### Step 1: Determine the value of \([x]\)
In the interval \(2 < x < 3\), the greatest integer function \([x]\) takes the value \(2\). Therefore, we can simplify the function:

\[ f(x) = \sin\left(\frac{\pi}{3} \cdot 2 - x^2\right) = \sin\left(\frac{2\pi}{3} - x^2\right) \]

### Step 2: Differentiate \(f(x)\)
Next, we differentiate \(f(x)\) with respect to \(x\):

\[ f'(x) = \frac{d}{dx} \left(\sin\left(\frac{2\pi}{3} - x^2\right)\right) \]

Using the chain rule, we have:

\[ f'(x) = \cos\left(\frac{2\pi}{3} - x^2\right) \cdot \frac{d}{dx}\left(\frac{2\pi}{3} - x^2\right) = \cos\left(\frac{2\pi}{3} - x^2\right) \cdot (-2x) \]

Thus, we can write:

\[ f'(x) = -2x \cos\left(\frac{2\pi}{3} - x^2\right) \]

### Step 3: Evaluate \(f'\left(\sqrt{\frac{\pi}{3}}\right)\)
Now, we need to evaluate \(f'\) at \(x = \sqrt{\frac{\pi}{3}}\):

\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cos\left(\frac{2\pi}{3} - \left(\sqrt{\frac{\pi}{3}}\right)^2\right) \]

Calculating \(\left(\sqrt{\frac{\pi}{3}}\right)^2\):

\[ \left(\sqrt{\frac{\pi}{3}}\right)^2 = \frac{\pi}{3} \]

Substituting this back into the cosine function:

\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cos\left(\frac{2\pi}{3} - \frac{\pi}{3}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cos\left(\frac{\pi}{3}\right) \]

### Step 4: Calculate \(\cos\left(\frac{\pi}{3}\right)\)
We know that:

\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

Substituting this value:

\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -2\left(\sqrt{\frac{\pi}{3}}\right) \cdot \frac{1}{2} \]

This simplifies to:

\[ f'\left(\sqrt{\frac{\pi}{3}}\right) = -\sqrt{\frac{\pi}{3}} \]

### Final Answer
Thus, the value of \( f'\left(\sqrt{\frac{\pi}{3}}\right) \) is:

\[ -\sqrt{\frac{\pi}{3}} \]