On which of the following intervals is the function f given by
`f(x)=x^(100)+sinx-1` strictly decreasing?

Options

A. (0, 1)
B. `(pi/2, pi)`
C. `(0, pi/2)`
D. None of these

Answer

To determine the intervals on which the function \( f(x) = x^{100} + \sin x - 1 \) is strictly decreasing, we need to analyze the derivative \( f'(x) \) and find where it is less than zero.

### Step-by-Step Solution:

1. **Find the derivative of the function:**
\[ f'(x) = \frac{d}{dx}(x^{100}) + \frac{d}{dx}(\sin x) - \frac{d}{dx}(1) \]
\[ f'(x) = 100x^{99} + \cos x \]

**Hint:** The derivative helps us determine the behavior of the function. A positive derivative indicates the function is increasing, while a negative derivative indicates it is decreasing.

2. **Set the derivative less than zero for strict decrease:**
\[ f'(x) < 0 \implies 100x^{99} + \cos x < 0 \]

**Hint:** We need to find the values of \( x \) for which this inequality holds true.

3. **Analyze the components of the inequality:**
- The term \( 100x^{99} \) is always non-negative for \( x \geq 0 \).
- The term \( \cos x \) oscillates between -1 and 1.

**Hint:** Since \( 100x^{99} \) grows very large as \( x \) increases, we need to find intervals where \( \cos x \) can potentially offset this term.

4. **Evaluate specific intervals:**
- **Interval A: \( [0, 1] \)**
- \( f'(0) = 100 \cdot 0^{99} + \cos(0) = 1 \) (positive)
- \( f'(1) = 100 \cdot 1^{99} + \cos(1) > 0 \) (since \( \cos(1) \) is positive)
- Conclusion: \( f'(x) > 0 \) in this interval (not decreasing).

- **Interval B: \( [\frac{\pi}{2}, \pi] \)**
- \( f'(\frac{\pi}{2}) = 100 \cdot (\frac{\pi}{2})^{99} + \cos(\frac{\pi}{2}) = 100 \cdot (\frac{\pi}{2})^{99} > 0 \)
- \( f'(\pi) = 100 \cdot \pi^{99} + \cos(\pi) = 100 \cdot \pi^{99} - 1 \) (still positive)
- Conclusion: \( f'(x) > 0 \) in this interval (not decreasing).

- **Interval C: \( [0, \frac{\pi}{2}] \)**
- \( f'(0) = 1 \) (positive)
- \( f'(\frac{\pi}{2}) > 0 \) (as calculated above)
- Conclusion: \( f'(x) > 0 \) in this interval (not decreasing).

- **Interval D: None of these**
- Since all intervals A, B, and C are increasing, the only option left is D.

5. **Final conclusion:**
The function \( f(x) = x^{100} + \sin x - 1 \) is not strictly decreasing in any of the given intervals. Therefore, the answer is:
\[ \text{Option D: None of these} \]