The function f(x) = sin x - x cos `pi` is

Options

A. maximum or minimum for all integral multiple of `pi`
B. maximum if x is an odd positive or even negative integral multiple of `pi`
C. minimum of x is an even positive or odd negative integral multiple of `pi`
D. None of these

Answer

To find the maxima and minima of the function \( f(x) = \sin x - x \cos x \), we will follow these steps:

### Step 1: Differentiate the function
We first need to find the first derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(\sin x) - \frac{d}{dx}(x \cos x) \]

Using the product rule for the second term:

\[ \frac{d}{dx}(x \cos x) = \cos x - x \sin x \]

Thus, the derivative becomes:

\[ f'(x) = \cos x - (\cos x - x \sin x) = x \sin x \]

### Step 2: Set the derivative to zero
To find the critical points, we set the first derivative equal to zero:

\[ f'(x) = x \sin x = 0 \]

This equation holds true when either \( x = 0 \) or \( \sin x = 0 \).

### Step 3: Solve for critical points
1. From \( x = 0 \), we have one critical point.
2. From \( \sin x = 0 \), we get \( x = n\pi \), where \( n \) is any integer.

Thus, the critical points are:

\[ x = 0, \, n\pi \quad (n \in \mathbb{Z}) \]

### Step 4: Determine the nature of critical points
To determine whether these critical points are maxima or minima, we can use the second derivative test. We need to find the second derivative \( f''(x) \).

### Step 5: Differentiate again
We differentiate \( f'(x) = x \sin x \):

\[ f''(x) = \frac{d}{dx}(x \sin x) = \sin x + x \cos x \]

### Step 6: Evaluate the second derivative at critical points
1. For \( x = 0 \):

\[ f''(0) = \sin(0) + 0 \cdot \cos(0) = 0 \]

Since \( f''(0) = 0 \), we cannot conclude anything about the nature of this critical point from the second derivative test.

2. For \( x = n\pi \):

\[ f''(n\pi) = \sin(n\pi) + n\pi \cos(n\pi) = 0 + n\pi (-1)^n = n\pi (-1)^n \]

- If \( n \) is even, \( f''(n\pi) > 0 \) (minimum).
- If \( n \) is odd, \( f''(n\pi) < 0 \) (maximum).

### Conclusion
- At \( x = 0 \), we need further analysis (potential inflection point).
- At \( x = n\pi \):
- Minima at even \( n \) (e.g., \( n = 0, 2, 4, \ldots \))
- Maxima at odd \( n \) (e.g., \( n = 1, 3, 5, \ldots \))