The right hand derivative of `f(x)=[x]t a npix a tx=7` is (where [.] denotes the greatest integer function) `0` b. `7pi` c. `-7pi` d. none of these

Options

A. 0
B. `7pi`
C. `-7pi`
D. None of these

Explanation

To find the right-hand derivative of the function \( f(x) = [x] \cdot 10 \cdot (x \cdot \pi) \) at \( x = 7 \), we will follow these steps:

### Step 1: Define the Right-Hand Derivative
The right-hand derivative of \( f \) at \( x = 7 \) is given by:
\[ f'(7^+) = \lim_{h \to 0^+} \frac{f(7 + h) - f(7)}{h} \]

### Step 2: Evaluate \( f(7) \)
First, we need to find \( f(7) \):
\[ f(7) = [7] \cdot 10 \cdot (7 \cdot \pi) = 7 \cdot 10 \cdot (7 \cdot \pi) = 70 \cdot 7 \cdot \pi = 490 \pi \]

### Step 3: Evaluate \( f(7 + h) \)
Next, we evaluate \( f(7 + h) \):
\[ f(7 + h) = [7 + h] \cdot 10 \cdot ((7 + h) \cdot \pi) \]
For small \( h \), \( [7 + h] = 7 \) (since \( h \) is positive and small enough that it does not increase the integer part). Thus:
\[ f(7 + h) = 7 \cdot 10 \cdot ((7 + h) \cdot \pi) = 7 \cdot 10 \cdot (7\pi + h\pi) = 70 \cdot (7\pi + h\pi) = 490\pi + 70h\pi \]

### Step 4: Substitute into the Derivative Formula
Now we substitute \( f(7 + h) \) and \( f(7) \) into the derivative formula:
\[ f'(7^+) = \lim_{h \to 0^+} \frac{(490\pi + 70h\pi) - 490\pi}{h} \]
This simplifies to:
\[ f'(7^+) = \lim_{h \to 0^+} \frac{70h\pi}{h} = \lim_{h \to 0^+} 70\pi = 70\pi \]

### Step 5: Conclusion
Thus, the right-hand derivative of \( f(x) \) at \( x = 7 \) is:
\[ f'(7^+) = 70\pi \]

### Final Answer
The answer is \( 70\pi \), which is not listed in the options provided (0, \( 7\pi \), \( -7\pi \), none of these). Therefore, the correct answer is **d. none of these**.