Given, `f(x)=abs(x-1), x in R`. Find Lf'(1)

Options

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

Answer

To find the left-hand derivative of the function \( f(x) = |x - 1| \) at \( x = 1 \), we will follow these steps:

### Step 1: Define the left-hand derivative
The left-hand derivative of a function \( f \) at a point \( a \) is defined as:
\[ Lf'(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} \]
In our case, we want to find \( Lf'(1) \).

### Step 2: Substitute into the definition
Substituting \( a = 1 \) into the definition gives us:
\[ Lf'(1) = \lim_{h \to 0^-} \frac{f(1 + h) - f(1)}{h} \]

### Step 3: Calculate \( f(1) \)
First, we need to find \( f(1) \):
\[ f(1) = |1 - 1| = |0| = 0 \]

### Step 4: Calculate \( f(1 + h) \)
Next, we calculate \( f(1 + h) \) for \( h < 0 \) (since we are approaching from the left):
\[ f(1 + h) = |(1 + h) - 1| = |h| \]
Since \( h \) is negative, \( |h| = -h \).

### Step 5: Substitute back into the limit
Now we substitute \( f(1 + h) \) and \( f(1) \) into the limit:
\[ Lf'(1) = \lim_{h \to 0^-} \frac{|h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} \]

### Step 6: Simplify the expression
This simplifies to:
\[ Lf'(1) = \lim_{h \to 0^-} -1 = -1 \]

### Conclusion
Thus, the left-hand derivative of \( f \) at \( x = 1 \) is:
\[ Lf'(1) = -1 \]