Let `f(x)=(x^(2)-3x+2)|(x^(3)-6x^(2)+11x-6)|+|sin(x+(pi)/(4))|` Number of points at which the function `f(x)` is non-differentiable in `[0,2 pi],` is

Options

A. 5
B. 4
C. 3
D. 2

Answer

To determine the number of points at which the function \( f(x) = (x^2 - 3x + 2) |(x^3 - 6x^2 + 11x - 6)| + |\sin(x + \frac{\pi}{4})| \) is non-differentiable in the interval \([0, 2\pi]\), we need to analyze the components of the function that can lead to non-differentiability.

### Step 1: Identify the critical points of \( x^2 - 3x + 2 \)

The expression \( x^2 - 3x + 2 \) can be factored as:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]
This quadratic is zero at \( x = 1 \) and \( x = 2 \).

### Step 2: Identify the critical points of \( x^3 - 6x^2 + 11x - 6 \)

Next, we need to find the roots of \( x^3 - 6x^2 + 11x - 6 \). We can use the Rational Root Theorem or synthetic division to find a root. Testing \( x = 1 \):
\[ 1^3 - 6(1^2) + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \]
So, \( x = 1 \) is a root. Now we can factor \( x^3 - 6x^2 + 11x - 6 \) as \( (x - 1)(x^2 - 5x + 6) \).

Next, we factor \( x^2 - 5x + 6 \):
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]
Thus, the complete factorization is:
\[ x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3) \]
The roots are \( x = 1, 2, 3 \).

### Step 3: Identify the critical points of \( |\sin(x + \frac{\pi}{4})| \)

The function \( |\sin(x + \frac{\pi}{4})| \) is non-differentiable at points where \( \sin(x + \frac{\pi}{4}) = 0 \). This occurs when:
\[ x + \frac{\pi}{4} = n\pi \quad \text{for } n \in \mathbb{Z} \]
Solving for \( x \):
\[ x = n\pi - \frac{\pi}{4} \]
For \( n = 0, 1, 2, \ldots \) within the interval \([0, 2\pi]\):
- For \( n = 0 \): \( x = -\frac{\pi}{4} \) (not in the interval)
- For \( n = 1 \): \( x = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \)
- For \( n = 2 \): \( x = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \)

### Step 4: Compile all non-differentiable points

Now, we compile all the points where \( f(x) \) is non-differentiable:
1. From \( x^2 - 3x + 2 \): \( x = 1, 2 \)
2. From \( |(x^3 - 6x^2 + 11x - 6)| \): \( x = 1, 2, 3 \)
3. From \( |\sin(x + \frac{\pi}{4})| \): \( x = \frac{3\pi}{4}, \frac{7\pi}{4} \)

### Step 5: List unique points

The unique points of non-differentiability in the interval \([0, 2\pi]\) are:
- \( x = 1 \)
- \( x = 2 \)
- \( x = 3 \)
- \( x = \frac{3\pi}{4} \)
- \( x = \frac{7\pi}{4} \)

### Conclusion

Thus, the total number of points at which the function \( f(x) \) is non-differentiable in the interval \([0, 2\pi]\) is **5**.