If `f(x)=|ax-b|+c|x|` is stricly increasing at atleast one point of non differentiability of the function where

Options

A. a. `c gt a`
B. b. `a gt c`
C. c. `b gt a +c`
D. d. `a =b`

Answer

To determine if the function \( f(x) = |ax - b| + c|x| \) is strictly increasing at least at one point of non-differentiability, we need to analyze the function based on the values of \( a \), \( b \), and \( c \).

### Step 1: Identify Points of Non-Differentiability
The function \( f(x) \) consists of absolute value expressions, which can cause non-differentiability at certain points. The points of non-differentiability occur where the expressions inside the absolute values equal zero:
1. \( ax - b = 0 \) → \( x = \frac{b}{a} \) (provided \( a \neq 0 \))
2. \( x = 0 \)

Thus, the points of non-differentiability are \( x = 0 \) and \( x = \frac{b}{a} \).

### Step 2: Analyze the Function in Different Intervals
We will analyze the function in the intervals defined by the points of non-differentiability.

1. **For \( x < 0 \)**:
- Here, \( ax - b < 0 \) and \( x < 0 \).
- Therefore, \( f(x) = -(ax - b) - cx = -ax + b - cx = (-a - c)x + b \).

2. **For \( 0 \leq x < \frac{b}{a} \)**:
- Here, \( ax - b < 0 \) and \( x \geq 0 \).
- Therefore, \( f(x) = -(ax - b) + cx = -ax + b + cx = (-a + c)x + b \).

3. **For \( x \geq \frac{b}{a} \)**:
- Here, \( ax - b \geq 0 \) and \( x \geq 0 \).
- Therefore, \( f(x) = (ax - b) + cx = ax - b + cx = (a + c)x - b \).

### Step 3: Determine the Slope in Each Interval
To check if \( f(x) \) is strictly increasing, we need to find the derivative in each interval:

1. **For \( x < 0 \)**:
- The slope is \( f'(x) = -a - c \).

2. **For \( 0 \leq x < \frac{b}{a} \)**:
- The slope is \( f'(x) = -a + c \).

3. **For \( x \geq \frac{b}{a} \)**:
- The slope is \( f'(x) = a + c \).

### Step 4: Conditions for Strictly Increasing
For \( f(x) \) to be strictly increasing at least at one point of non-differentiability, we need at least one of the slopes to be positive:

1. **At \( x = 0 \)**:
- The slope is \( -a + c \). For this to be positive, we need \( c > a \).

2. **At \( x = \frac{b}{a} \)**:
- The slope is \( a + c \). For this to be positive, we need \( a + c > 0 \).

### Conclusion
From the analysis, we conclude that:
- If \( c > a \), then \( f(x) \) is strictly increasing at \( x = 0 \).
- If \( a + c > 0 \), then \( f(x) \) is strictly increasing at \( x = \frac{b}{a} \).

Thus, the function \( f(x) \) is strictly increasing at least at one point of non-differentiability if \( c > a \).