The number of points where `f(x)= abs(2x-1) -3abs(x+2)+abs(x^2+x-2) , ninR` is not differentiable is

Answer

To find the number of points where the function \( f(x) = |2x - 1| - 3|x + 2| + |x^2 + x - 2| \) is not differentiable, we need to identify the points where the absolute value expressions change their behavior, which typically occurs at the points where the expressions inside the absolute values equal zero.

### Step 1: Identify the points where each absolute value expression is zero.

1. For \( |2x - 1| = 0 \):
\[ 2x - 1 = 0 \implies x = \frac{1}{2} \]

2. For \( |x + 2| = 0 \):
\[ x + 2 = 0 \implies x = -2 \]

3. For \( |x^2 + x - 2| = 0 \):
- Factor the quadratic:
\[ x^2 + x - 2 = (x + 2)(x - 1) = 0 \]
- This gives:
\[ x = -2 \quad \text{and} \quad x = 1 \]

### Step 2: List all points where the function may not be differentiable.

From the above calculations, we have the following points:
- \( x = \frac{1}{2} \)
- \( x = -2 \)
- \( x = 1 \)

### Step 3: Check the points for differentiability.

Next, we need to check the point \( x = -2 \) specifically, as it appears twice (from \( |x + 2| \) and \( |x^2 + x - 2| \)).

To determine if \( f(x) \) is differentiable at \( x = -2 \), we need to compute the left-hand derivative and right-hand derivative at this point.

1. **Left-hand derivative at \( x = -2 \)**:
- For \( x < -2 \):
\[ f(x) = -(2x - 1) - 3(-x - 2) + -(x^2 + x - 2) \]
Simplifying this gives:
\[ f(x) = -2x + 1 + 3x + 6 - (x^2 + x - 2) = -x^2 + 2x + 5 \]
Thus, the derivative is:
\[ f'(x) = -2x + 2 \quad \text{for } x < -2 \]
Evaluating at \( x = -2 \):
\[ f'(-2) = -2(-2) + 2 = 4 + 2 = 6 \]

2. **Right-hand derivative at \( x = -2 \)**:
- For \( x > -2 \):
\[ f(x) = (2x - 1) - 3(x + 2) + (x^2 + x - 2) \]
Simplifying this gives:
\[ f(x) = 2x - 1 - 3x - 6 + x^2 + x - 2 = x^2 - 2x - 9 \]
Thus, the derivative is:
\[ f'(x) = 2x - 2 \quad \text{for } x > -2 \]
Evaluating at \( x = -2 \):
\[ f'(-2) = 2(-2) - 2 = -4 - 2 = -6 \]

### Step 4: Compare the left-hand and right-hand derivatives.

- Left-hand derivative at \( x = -2 \): \( 6 \)
- Right-hand derivative at \( x = -2 \): \( -6 \)

Since the left-hand and right-hand derivatives are not equal, \( f(x) \) is not differentiable at \( x = -2 \).

### Conclusion

The points where \( f(x) \) is not differentiable are:
- \( x = \frac{1}{2} \)
- \( x = -2 \)
- \( x = 1 \)

Thus, the total number of points where \( f(x) \) is not differentiable is **3**.

### Final Answer
The number of points where \( f(x) \) is not differentiable is **3**.