Let `f(x)= {{:(1+ sin x, x lt 0 ),(x^2-x+1, x ge 0 ):}`

Options

A. f has a local maximum at `x=0`
B. f has a local minimum at `x=0`
C. f is increasing everywhere
D. f is decreasing everywhere

Answer

To solve the problem step-by-step, we will analyze the function \( f(x) \) defined piecewise and find its derivative to determine the intervals of increase and decrease, as well as local maxima and minima.

### Step 1: Define the function
The function is given as:
\[ f(x) = \begin{cases} 1 + \sin x & \text{if } x < 0 \\ x^2 - x + 1 & \text{if } x \geq 0 \end{cases} \]

### Step 2: Find the derivative \( f'(x) \)
We need to find the derivative for both cases of the piecewise function.

1. For \( x < 0 \):
\[ f'(x) = \frac{d}{dx}(1 + \sin x) = \cos x \]

2. For \( x \geq 0 \):
\[ f'(x) = \frac{d}{dx}(x^2 - x + 1) = 2x - 1 \]

Thus, we have:
\[ f'(x) = \begin{cases} \cos x & \text{if } x < 0 \\ 2x - 1 & \text{if } x \geq 0 \end{cases} \]

### Step 3: Analyze \( f'(x) \) for \( x < 0 \)
For \( x < 0 \), we analyze \( f'(x) = \cos x \):
- \( \cos x \) is positive in the interval \( (-\frac{\pi}{2}, 0) \) and negative in \( (-\pi, -\frac{\pi}{2}) \).
- Therefore, \( f(x) \) is increasing in \( (-\frac{\pi}{2}, 0) \).

### Step 4: Analyze \( f'(x) \) for \( x \geq 0 \)
For \( x \geq 0 \), we analyze \( f'(x) = 2x - 1 \):
- Set \( f'(x) = 0 \) to find critical points:
\[ 2x - 1 = 0 \implies x = \frac{1}{2} \]
- For \( 0 \leq x < \frac{1}{2} \), \( f'(x) < 0 \) (decreasing).
- For \( x > \frac{1}{2} \), \( f'(x) > 0 \) (increasing).

### Step 5: Determine local maxima and minima
- At \( x = 0 \):
- \( f'(0) = 2(0) - 1 = -1 \) (decreasing).
- \( f(0) = 1 \).

- At \( x = \frac{1}{2} \):
- \( f'(\frac{1}{2}) = 0 \) (critical point).
- \( f(\frac{1}{2}) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{3}{4} \).

### Step 6: Summary of intervals
- \( f(x) \) is increasing in \( (-\frac{\pi}{2}, 0) \).
- \( f(x) \) is decreasing in \( [0, \frac{1}{2}) \).
- \( f(x) \) is increasing in \( (\frac{1}{2}, \infty) \).

### Step 7: Identify local maxima and minima
- Local maximum at \( x = 0 \) (since it changes from increasing to decreasing).
- Local minimum at \( x = \frac{1}{2} \) (since it changes from decreasing to increasing).

### Final Conclusion
The function \( f(x) \) has:
- A local maximum at \( x = 0 \).
- A local minimum at \( x = \frac{1}{2} \).