The function `f(x)=2|x|+|x+2|=||x|2|-2|x||` has a local minimum or a local maximum at `x=` `-2` (b) `-2/3` (c) 2 (d) `2/3`

Options

A. -2
B. `-2//3`
C. 2
D. `2//3`

Explanation

To determine whether the function \( f(x) = 2|x| + |x+2| \) has a local minimum or maximum at the given points, we will analyze the function step by step.

### Step 1: Identify the critical points
First, we need to find the points where the function changes its behavior. The absolute value functions can change at the points where their arguments are zero.

1. For \( |x| \), it changes at \( x = 0 \).
2. For \( |x + 2| \), it changes at \( x = -2 \).

Thus, the critical points to consider are \( x = -2 \) and \( x = 0 \).

### Step 2: Analyze the function in intervals
Next, we will analyze \( f(x) \) in the intervals determined by the critical points: \( (-\infty, -2) \), \( [-2, 0) \), and \( [0, \infty) \).

1. **Interval \( (-\infty, -2) \)**:
- Here, \( x < -2 \) implies \( |x| = -x \) and \( |x + 2| = -(x + 2) \).
- Therefore, \( f(x) = 2(-x) + (-(x + 2)) = -2x - x - 2 = -3x - 2 \).

2. **Interval \( [-2, 0) \)**:
- Here, \( -2 \leq x < 0 \) implies \( |x| = -x \) and \( |x + 2| = x + 2 \).
- Therefore, \( f(x) = 2(-x) + (x + 2) = -2x + x + 2 = -x + 2 \).

3. **Interval \( [0, \infty) \)**:
- Here, \( x \geq 0 \) implies \( |x| = x \) and \( |x + 2| = x + 2 \).
- Therefore, \( f(x) = 2x + (x + 2) = 3x + 2 \).

### Step 3: Determine the behavior at critical points
Now we will evaluate the function at the critical points and check the behavior:

1. **At \( x = -2 \)**:
- From the left interval \( (-\infty, -2) \): \( f(-2) = -3(-2) - 2 = 6 - 2 = 4 \).
- From the right interval \( [-2, 0) \): \( f(-2) = -(-2) + 2 = 2 + 2 = 4 \).

2. **At \( x = 0 \)**:
- From the left interval \( [-2, 0) \): \( f(0) = -0 + 2 = 2 \).
- From the right interval \( [0, \infty) \): \( f(0) = 3(0) + 2 = 2 \).

### Step 4: Check for local minima or maxima
- At \( x = -2 \), \( f(x) \) changes from increasing to decreasing, indicating a local maximum.
- At \( x = 0 \), \( f(x) \) remains constant, indicating neither a local minimum nor a maximum.

### Conclusion
The function \( f(x) = 2|x| + |x + 2| \) has a local maximum at \( x = -2 \).

### Final Answer
The function has a local maximum at \( x = -2 \).

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