The function `|x^2 - 3x+2| + cos |x|` is not differentiableat `x=`

Options

A. `-1,2`
B. `-1,-2`
C. 1,2
D. `-2,1

Answer

To determine where the function \( f(x) = |x^2 - 3x + 2| + \cos |x| \) is not differentiable, we need to analyze the components of the function separately.

### Step 1: Identify the points of non-differentiability for \( |x^2 - 3x + 2| \)

First, we need to factor the quadratic expression inside the absolute value:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]
The roots of this equation are \( x = 1 \) and \( x = 2 \). The function \( |x^2 - 3x + 2| \) will change its behavior at these points.

### Step 2: Determine the intervals for the absolute value

Next, we analyze the sign of \( x^2 - 3x + 2 \) in the intervals defined by the roots:
- For \( x < 1 \): \( x^2 - 3x + 2 > 0 \) (the expression is positive)
- For \( 1 < x < 2 \): \( x^2 - 3x + 2 < 0 \) (the expression is negative)
- For \( x > 2 \): \( x^2 - 3x + 2 > 0 \) (the expression is positive)

Thus, we can express \( |x^2 - 3x + 2| \) as:
\[ |x^2 - 3x + 2| = \begin{cases} x^2 - 3x + 2 & \text{if } x < 1 \\ -(x^2 - 3x + 2) & \text{if } 1 \leq x < 2 \\ x^2 - 3x + 2 & \text{if } x \geq 2 \end{cases} \]

### Step 3: Check differentiability at the points \( x = 1 \) and \( x = 2 \)

To check differentiability at \( x = 1 \):
- The left-hand derivative as \( x \) approaches 1:
\[ f'(1^-) = \frac{d}{dx}(x^2 - 3x + 2) \bigg|_{x=1} = 2(1) - 3 = -1 \]
- The right-hand derivative as \( x \) approaches 1:
\[ f'(1^+) = \frac{d}{dx}(-(x^2 - 3x + 2)) \bigg|_{x=1} = -[2(1) - 3] = 1 \]
Since the left-hand and right-hand derivatives are not equal, \( f(x) \) is not differentiable at \( x = 1 \).

To check differentiability at \( x = 2 \):
- The left-hand derivative as \( x \) approaches 2:
\[ f'(2^-) = \frac{d}{dx}(-(x^2 - 3x + 2)) \bigg|_{x=2} = -[2(2) - 3] = -1 \]
- The right-hand derivative as \( x \) approaches 2:
\[ f'(2^+) = \frac{d}{dx}(x^2 - 3x + 2) \bigg|_{x=2} = 2(2) - 3 = 1 \]
Again, the left-hand and right-hand derivatives are not equal, so \( f(x) \) is not differentiable at \( x = 2 \).

### Conclusion

The function \( f(x) = |x^2 - 3x + 2| + \cos |x| \) is not differentiable at \( x = 1 \) and \( x = 2 \).

### Final Answer
The function is not differentiable at \( x = 1 \) and \( x = 2 \).