If `f(x)=cos{(pi)/(2)[x]-x^(3)},1ltxlt2`, and [x] denotes the greatest integer less than or equal to x, then the value of
`f'(root(3)((pi)/(2)))`, is

Options

A. 0
B. `3((pi)/(2))^(2//3)`
C. `-3((pi)/(2))^(2//3)`
D. none of these

Explanation

To solve the problem step by step, we need to find the derivative of the function \( f(x) = \cos\left(\frac{\pi}{2} [x] - x^3\right) \) for \( 1 < x < 2 \) and evaluate it at \( x = \sqrt[3]{\frac{\pi}{2}} \).

### Step 1: Identify the function and the interval
Given:
\[ f(x) = \cos\left(\frac{\pi}{2} [x] - x^3\right) \]
where \( [x] \) is the greatest integer less than or equal to \( x \). Since \( 1 < x < 2 \), we have \( [x] = 1 \).

### Step 2: Simplify the function
Substituting \( [x] = 1 \) into the function:
\[ f(x) = \cos\left(\frac{\pi}{2} \cdot 1 - x^3\right) = \cos\left(\frac{\pi}{2} - x^3\right) \]

### Step 3: Use the cosine identity
Using the identity \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta) \):
\[ f(x) = \sin(x^3) \]

### Step 4: Differentiate the function
Now, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx} \sin(x^3) = \cos(x^3) \cdot \frac{d}{dx}(x^3) = \cos(x^3) \cdot 3x^2 \]

### Step 5: Evaluate the derivative at \( x = \sqrt[3]{\frac{\pi}{2}} \)
Now, we need to evaluate \( f'(\sqrt[3]{\frac{\pi}{2}}) \):
\[ f'\left(\sqrt[3]{\frac{\pi}{2}}\right) = \cos\left(\left(\sqrt[3]{\frac{\pi}{2}}\right)^3\right) \cdot 3\left(\sqrt[3]{\frac{\pi}{2}}\right)^2 \]
Since \( \left(\sqrt[3]{\frac{\pi}{2}}\right)^3 = \frac{\pi}{2} \):
\[ f'\left(\sqrt[3]{\frac{\pi}{2}}\right) = \cos\left(\frac{\pi}{2}\right) \cdot 3\left(\sqrt[3]{\frac{\pi}{2}}\right)^2 \]

### Step 6: Calculate \( \cos\left(\frac{\pi}{2}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]
Thus:
\[ f'\left(\sqrt[3]{\frac{\pi}{2}}\right) = 0 \cdot 3\left(\sqrt[3]{\frac{\pi}{2}}\right)^2 = 0 \]

### Final Answer
The value of \( f'\left(\sqrt[3]{\frac{\pi}{2}}\right) \) is \( 0 \).

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