STATEMENT-1: `f(x)=|x-1|+|x+2|+|x-3|` has a local minima . and STATEMENT-2 : Any differentiable function f(x) may have a local maxima or minima if f'(x)=0 at some points

Options

A. Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement-1 .
B. Statement-1 is True , Statement-2 is True , Statement-2 is NOT a correct explanation for Statement-1 .
C. Statement-1 is True , Statement-2 is False
D. Statement-1 is False , Statement-2 is True

Answer

To solve the problem, we need to analyze the function \( f(x) = |x - 1| + |x + 2| + |x - 3| \) and determine whether it has a local minimum, and also evaluate the second statement regarding differentiable functions.

### Step 1: Identify the critical points of the function
The function \( f(x) \) consists of three absolute value terms. The critical points occur where each absolute value term changes, which is at the points where the expressions inside the absolute values equal zero.

1. \( |x - 1| \) changes at \( x = 1 \)
2. \( |x + 2| \) changes at \( x = -2 \)
3. \( |x - 3| \) changes at \( x = 3 \)

Thus, the critical points are \( x = -2, 1, 3 \).

### Step 2: Analyze the intervals
Next, we will analyze the function in the intervals defined by these critical points:

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

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

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

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

### Step 3: Evaluate the function at critical points
Now, we will evaluate \( f(x) \) at the critical points:

- \( f(-2) = | -2 - 1 | + | -2 + 2 | + | -2 - 3 | = 3 + 0 + 5 = 8 \)
- \( f(1) = | 1 - 1 | + | 1 + 2 | + | 1 - 3 | = 0 + 3 + 2 = 5 \)
- \( f(3) = | 3 - 1 | + | 3 + 2 | + | 3 - 3 | = 2 + 5 + 0 = 7 \)

### Step 4: Determine local minima and maxima
From the evaluations, we see:
- \( f(-2) = 8 \)
- \( f(1) = 5 \) (local minimum)
- \( f(3) = 7 \)

Thus, \( f(x) \) has a local minimum at \( x = 1 \).

### Conclusion for Statement 1
**Statement 1** is correct: \( f(x) \) has a local minimum.

### Step 5: Analyze Statement 2
**Statement 2** claims that any differentiable function \( f(x) \) may have a local maxima or minima if \( f'(x) = 0 \) at some points. This is true, as critical points where the derivative is zero can indicate local maxima, minima, or saddle points.

### Conclusion for Statement 2
**Statement 2** is also correct.

### Final Answer
Both statements are correct.