Consider the function `f(x)={{:(xsin(pi)/(x),"for"xgt0),(0,"for"x=0):}`
Then, the number of points in `(0,1)` where the derivative f'(x) vanishes is

Options

A. 0
B. 1
C. 2
D. infinite

Answer

To solve the problem, we need to analyze the function \( f(x) = x \sin\left(\frac{\pi}{x}\right) \) for \( x > 0 \) and \( f(0) = 0 \). We are tasked with finding the number of points in the interval \( (0, 1) \) where the derivative \( f'(x) \) vanishes.

### Step 1: Find the Derivative \( f'(x) \)

To find \( f'(x) \), we will use the product rule. The function can be expressed as:

\[ f(x) = x \sin\left(\frac{\pi}{x}\right) \]

Using the product rule, we have:

\[ f'(x) = \sin\left(\frac{\pi}{x}\right) + x \cdot \frac{d}{dx}\left(\sin\left(\frac{\pi}{x}\right)\right) \]

Now, we need to differentiate \( \sin\left(\frac{\pi}{x}\right) \):

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

Substituting this back into the derivative, we get:

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

### Step 2: Set the Derivative to Zero

We need to find where \( f'(x) = 0 \):

\[ \sin\left(\frac{\pi}{x}\right) - \frac{\pi}{x} \cos\left(\frac{\pi}{x}\right) = 0 \]

This can be rearranged to:

\[ \sin\left(\frac{\pi}{x}\right) = \frac{\pi}{x} \cos\left(\frac{\pi}{x}\right) \]

### Step 3: Analyze the Equation

This equation can be analyzed by considering the values of \( \frac{\pi}{x} \). The sine function equals zero at integer multiples of \( \pi \):

\[ \frac{\pi}{x} = k\pi \quad \text{for } k \in \mathbb{Z} \]

This gives us:

\[ x = \frac{1}{k} \quad \text{for } k = 1, 2, 3, \ldots \]

### Step 4: Determine Valid \( k \) Values in \( (0, 1) \)

We need to find the values of \( k \) such that \( x = \frac{1}{k} \) lies in the interval \( (0, 1) \). This is true for all natural numbers \( k \geq 1 \).

Thus, the valid values of \( k \) are \( 1, 2, 3, \ldots \), which correspond to the points:

\[ x = 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \]

### Step 5: Count the Points

Since \( k \) can take any natural number value, there are infinitely many points \( x = \frac{1}{k} \) in the interval \( (0, 1) \) where \( f'(x) = 0 \).

### Conclusion

The number of points in \( (0, 1) \) where the derivative \( f'(x) \) vanishes is **infinite**.