Consider the function `f(x)=(x^(3)-x)|x^(2)-6x+5|, AA x in R`, then f(x) is

Options

A. discontinuous at x = 1
B. discontinuous at x = 5
C. non differentiable at x = 1
D. non differentiable at x = 5

Answer

To analyze the function \( f(x) = (x^3 - x) |x^2 - 6x + 5| \), we will follow these steps:

### Step 1: Factor the expression inside the modulus
First, we need to factor the quadratic expression \( x^2 - 6x + 5 \).

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

### Step 2: Determine the intervals where the expression is positive or negative
Next, we need to find the intervals where \( |x^2 - 6x + 5| \) is positive or negative. The roots of the quadratic are \( x = 1 \) and \( x = 5 \). We will analyze the sign of \( x^2 - 6x + 5 \) in the intervals:

1. \( (-\infty, 1) \)
2. \( (1, 5) \)
3. \( (5, \infty) \)

- For \( x < 1 \) (e.g., \( x = 0 \)):
\[ 0^2 - 6(0) + 5 = 5 > 0 \quad \Rightarrow \quad |x^2 - 6x + 5| = x^2 - 6x + 5 \]

- For \( 1 < x < 5 \) (e.g., \( x = 3 \)):
\[ 3^2 - 6(3) + 5 = 9 - 18 + 5 = -4 < 0 \quad \Rightarrow \quad |x^2 - 6x + 5| = -(x^2 - 6x + 5) \]

- For \( x > 5 \) (e.g., \( x = 6 \)):
\[ 6^2 - 6(6) + 5 = 36 - 36 + 5 = 5 > 0 \quad \Rightarrow \quad |x^2 - 6x + 5| = x^2 - 6x + 5 \]

### Step 3: Rewrite the function based on the intervals
Now we can rewrite \( f(x) \) based on the intervals:

- For \( x < 1 \):
\[ f(x) = (x^3 - x)(x^2 - 6x + 5) \]

- For \( 1 < x < 5 \):
\[ f(x) = (x^3 - x)(-(x^2 - 6x + 5)) = -(x^3 - x)(x^2 - 6x + 5) \]

- For \( x > 5 \):
\[ f(x) = (x^3 - x)(x^2 - 6x + 5) \]

### Step 4: Identify points of discontinuity and non-differentiability
The function \( f(x) \) is a polynomial multiplied by a piecewise function due to the modulus.

- **Discontinuity**: A polynomial function is continuous everywhere, so we check the points where the modulus changes, which are \( x = 1 \) and \( x = 5 \).

- **Non-differentiability**: The function may be non-differentiable at points where the modulus changes, i.e., at \( x = 1 \) and \( x = 5 \).

### Conclusion
- **Points of discontinuity**: 0
- **Points of non-differentiability**: 2 (at \( x = 1 \) and \( x = 5 \))