Let `f : R to R` be differentiable at ` c in R and f(c ) = 0` . If g(x) = |f(x) |, then at x = c, g is

Options

A. not differentiable
B. differentiable if `f'(c ) ne 0 `
C. not differentiable if ` f'(c ) = 0 `
D. differentiable if ` f'( c ) = 0 `

Explanation

To determine whether the function \( g(x) = |f(x)| \) is differentiable at \( x = c \), given that \( f \) is differentiable at \( c \) and \( f(c) = 0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that \( f: \mathbb{R} \to \mathbb{R} \) is differentiable at \( c \).
- We have \( f(c) = 0 \).
- We need to analyze the function \( g(x) = |f(x)| \) at \( x = c \).

2. **Define the Derivative of \( g \)**:
- The derivative of \( g \) at \( c \) is defined as:
\[ g'(c) = \lim_{h \to 0} \frac{g(c+h) - g(c)}{h} \]
- Since \( g(c) = |f(c)| = |0| = 0 \), we can rewrite this as:
\[ g'(c) = \lim_{h \to 0} \frac{|f(c+h)| - 0}{h} = \lim_{h \to 0} \frac{|f(c+h)|}{h} \]

3. **Consider the Behavior of \( f \)**:
- Since \( f \) is differentiable at \( c \), we can apply the definition of the derivative:
\[ f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} = \lim_{h \to 0} \frac{f(c+h)}{h} \]
- This means that as \( h \) approaches \( 0 \), \( f(c+h) \) approaches \( f(c) = 0 \).

4. **Evaluate the Limit for \( g'(c) \)**:
- We need to analyze \( |f(c+h)| \) as \( h \) approaches \( 0 \):
- If \( f'(c) > 0 \), then \( f(c+h) \) will be positive for small \( h \), and thus \( |f(c+h)| = f(c+h) \).
- If \( f'(c) < 0 \), then \( f(c+h) \) will be negative for small \( h \), and thus \( |f(c+h)| = -f(c+h) \).
- If \( f'(c) = 0 \), then \( f(c+h) \) approaches \( 0 \) from both sides.

5. **Final Evaluation**:
- If \( f'(c) \neq 0 \), then \( g'(c) \) will be equal to \( |f'(c)| \), and thus \( g \) is differentiable at \( c \).
- If \( f'(c) = 0 \), we need to check the left-hand and right-hand derivatives:
- Both left-hand and right-hand derivatives will equal \( 0 \), thus \( g'(c) \) exists and is equal to \( 0 \).

### Conclusion:
- Therefore, we conclude that \( g(x) \) is differentiable at \( x = c \) if \( f'(c) = 0 \) and also if \( f'(c) \neq 0 \). Hence, the correct option is:
- **Differentiable if \( f'(c) = 0 \)**.