if `f(x)=(x^2-4)|(x^3-6x^2+11x-6)|+x/(1+|x|)`then set of points at which the function if non differentiable is

Options

A. {-2,2,1,3}
B. {-2,0,3}
C. {-2,2,0}
D. {1,3}

Answer

To determine the set of points at which the function \( f(x) = (x^2 - 4) |x^3 - 6x^2 + 11x - 6| + \frac{x}{1 + |x|} \) is non-differentiable, we will follow these steps:

### Step 1: Identify the components of the function
The function consists of two main parts:
1. \( (x^2 - 4) |x^3 - 6x^2 + 11x - 6| \)
2. \( \frac{x}{1 + |x|} \)

### Step 2: Analyze the absolute value function
The expression \( |x^3 - 6x^2 + 11x - 6| \) is non-differentiable at points where \( x^3 - 6x^2 + 11x - 6 = 0 \). We need to find the roots of the polynomial:

\[ x^3 - 6x^2 + 11x - 6 = 0 \]

### Step 3: Factor the polynomial
Using the Rational Root Theorem or synthetic division, we can find the roots. Testing \( x = 1 \):

\[ 1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \]

Thus, \( x = 1 \) is a root. We can factor the polynomial as follows:

\[ x^3 - 6x^2 + 11x - 6 = (x - 1)(x^2 - 5x + 6) \]

Now, we can factor \( x^2 - 5x + 6 \):

\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

Thus, we have:

\[ x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) \]

### Step 4: Identify critical points
The critical points where \( |x^3 - 6x^2 + 11x - 6| \) is non-differentiable are \( x = 1, 2, 3 \).

### Step 5: Check the effect of \( (x^2 - 4) \)
The term \( (x^2 - 4) \) becomes zero at \( x = 2 \) and \( x = -2 \). At \( x = 2 \), since \( (x^2 - 4) \) is multiplied by \( |x^3 - 6x^2 + 11x - 6| \), it neutralizes the non-differentiability at this point.

### Step 6: Analyze \( \frac{x}{1 + |x|} \)
The term \( \frac{x}{1 + |x|} \) is non-differentiable at \( x = 0 \) due to the absolute value in the denominator.

### Step 7: Compile the non-differentiable points
From our analysis, the points where \( f(x) \) is non-differentiable are:
- \( x = 1 \) (due to the absolute value)
- \( x = 3 \) (due to the absolute value)
- \( x = 0 \) (due to the absolute value in the denominator)

### Final Answer
The set of points at which the function is non-differentiable is:
\[ \{0, 1, 3\} \]