Let `g(x)=||x+2|-3|`. If a denotes the number of relative minima, b denotes the number of relative maxima and c denotes the product of the zeros. Then the value of `(a+2b-c)` is______.

Answer

To solve the problem, we need to analyze the function \( g(x) = ||x + 2| - 3| \) and determine the number of relative minima (a), the number of relative maxima (b), and the product of the zeros (c). Finally, we will calculate \( a + 2b - c \).

### Step 1: Understand the function
The function \( g(x) \) is composed of nested absolute values. First, we need to break down the function into simpler parts.

1. **Inner Absolute Value**:
\[ h(x) = |x + 2| \]
This function equals \( x + 2 \) for \( x \geq -2 \) and \( -(x + 2) \) for \( x < -2 \).

2. **Next Step**:
\[ k(x) = |h(x) - 3| = ||x + 2| - 3| \]
This function will change depending on the value of \( h(x) \).

### Step 2: Find critical points
To find the critical points, we need to analyze where \( g(x) \) changes its behavior:

1. **Identify points where \( |x + 2| = 3 \)**:
- \( x + 2 = 3 \) gives \( x = 1 \)
- \( x + 2 = -3 \) gives \( x = -5 \)

2. **Identify the points where \( |x + 2| \) changes**:
- At \( x = -2 \), the expression \( |x + 2| \) changes.

### Step 3: Evaluate the function in intervals
We will evaluate \( g(x) \) in the intervals defined by the critical points \( -5, -2, \) and \( 1 \).

1. **Interval \( (-\infty, -5) \)**:
\[ g(x) = |-(x + 2) - 3| = |-(x + 5)| = x + 5 \]

2. **Interval \( [-5, -2) \)**:
\[ g(x) = |-(x + 2) - 3| = |-(x + 5)| = -x - 5 \]

3. **Interval \( [-2, 1) \)**:
\[ g(x) = |x + 2 - 3| = |x - 1| = -(x - 1) = -x + 1 \]

4. **Interval \( [1, \infty) \)**:
\[ g(x) = |x + 2 - 3| = |x - 1| = x - 1 \]

### Step 4: Identify relative maxima and minima
Now we can find the relative maxima and minima by evaluating the function at the critical points and checking the behavior around them.

1. **At \( x = -5 \)**:
- \( g(-5) = 0 \) (local minimum)

2. **At \( x = -2 \)**:
- \( g(-2) = 1 \) (local maximum)

3. **At \( x = 1 \)**:
- \( g(1) = 0 \) (local minimum)

### Step 5: Count relative maxima and minima
- **Relative minima (a)**: 2 (at \( x = -5 \) and \( x = 1 \))
- **Relative maxima (b)**: 1 (at \( x = -2 \))

### Step 6: Find the product of the zeros (c)
The zeros of \( g(x) \) are at \( x = -5 \) and \( x = 1 \):
\[ c = (-5) \times 1 = -5 \]

### Step 7: Calculate \( a + 2b - c \)
Now we can substitute the values into the expression:
\[ a + 2b - c = 2 + 2(1) - (-5) = 2 + 2 + 5 = 9 \]

### Final Answer
The value of \( a + 2b - c \) is \( \boxed{9} \).