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

Explanation

To solve the problem, we need to analyze the function \( g(x) = |f(x)| \) at the point \( x = c \), given that \( f(c) = 0 \) and \( f \) is differentiable at \( c \).

### Step-by-Step Solution:

1. **Understanding the Function**:
We know that \( g(x) = |f(x)| \). Since \( f(c) = 0 \), we can evaluate \( g(c) \):
\[ g(c) = |f(c)| = |0| = 0. \]

2. **Finding the Derivative of \( g \)**:
To find \( g'(c) \), we need to consider the definition of the derivative:
\[ g'(c) = \lim_{h \to 0} \frac{g(c+h) - g(c)}{h} = \lim_{h \to 0} \frac{|f(c+h)| - |f(c)|}{h} = \lim_{h \to 0} \frac{|f(c+h)| - 0}{h} = \lim_{h \to 0} \frac{|f(c+h)|}{h}. \]

3. **Evaluating the Limit from the Right**:
For \( h > 0 \) (right-hand limit):
\[ g'(c) = \lim_{h \to 0^+} \frac{|f(c+h)|}{h}. \]
Since \( f \) is differentiable at \( c \), we can express \( f(c+h) \) using the definition of the derivative:
\[ f(c+h) = f(c) + f'(c)h + o(h) = 0 + f'(c)h + o(h) = f'(c)h + o(h). \]
Therefore,
\[ |f(c+h)| = |f'(c)h + o(h)|. \]

4. **Taking the Limit**:
As \( h \to 0^+ \):
\[ g'(c) = \lim_{h \to 0^+} \frac{|f'(c)h + o(h)|}{h} = \lim_{h \to 0^+} \left| f'(c) + \frac{o(h)}{h} \right|. \]
Since \( \frac{o(h)}{h} \to 0 \) as \( h \to 0 \), we have:
\[ g'(c) = |f'(c)|. \]

5. **Evaluating the Limit from the Left**:
For \( h < 0 \) (left-hand limit):
\[ g'(c) = \lim_{h \to 0^-} \frac{|f(c+h)|}{h}. \]
Using a similar argument as above, we find:
\[ g'(c) = \lim_{h \to 0^-} \frac{|f'(c)h + o(h)|}{h} = \lim_{h \to 0^-} \left| f'(c) + \frac{o(h)}{h} \right| = |f'(c)|. \]

6. **Conclusion**:
Since both the right-hand limit and the left-hand limit yield \( |f'(c)| \), we conclude that:
\[ g'(c) = |f'(c)|. \]
Therefore, \( g \) is differentiable at \( c \) if \( f'(c) \neq 0 \) and is not differentiable if \( f'(c) = 0 \).

### Final Answer:
At \( x = c \), \( g \) is differentiable if \( f'(c) \neq 0 \) and not differentiable if \( f'(c) = 0 \).