Let `f(x)={{:(-2",",-3lexle0),(x-2",",0ltxle3):}andg(x)=f(|x|)+|f(x)|`
What is the value of the differential coefficient of g(x) at x=-2?

Options

A. `-1`
B. 0
C. 1
D. 2

Explanation

To find the differential coefficient of \( g(x) \) at \( x = -2 \), we will follow these steps:

### Step 1: Define the function \( f(x) \)
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} -2 & \text{if } -3 \leq x \leq 0 \\ x - 2 & \text{if } 0 < x \leq 3 \end{cases} \]

### Step 2: Determine \( f(|x|) \) and \( |f(x)| \)
To find \( g(x) \), we need to evaluate \( f(|x|) \) and \( |f(x)| \).

1. For \( |x| \) when \( x = -2 \):
\[ |x| = 2 \quad \Rightarrow \quad f(|x|) = f(2) = 2 - 2 = 0 \]

2. For \( f(x) \) when \( x = -2 \):
\[ f(-2) = -2 \quad \Rightarrow \quad |f(-2)| = |-2| = 2 \]

### Step 3: Calculate \( g(x) \)
Now we can find \( g(x) \):
\[ g(x) = f(|x|) + |f(x)| = f(2) + |f(-2)| = 0 + 2 = 2 \]

### Step 4: Differentiate \( g(x) \)
Since \( g(x) \) is a constant function (equal to 2) around \( x = -2 \), its derivative will be:
\[ g'(x) = 0 \quad \text{for all } x \]

### Step 5: Find the differential coefficient at \( x = -2 \)
Thus, the differential coefficient of \( g(x) \) at \( x = -2 \) is:
\[ g'(-2) = 0 \]

### Final Answer
The value of the differential coefficient of \( g(x) \) at \( x = -2 \) is \( \boxed{0} \).

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