`L e tf(x)={x+2,x<-1x^2,-1lt=x<1(x-2)^2,xgeq1` Then number of times `f^(prime)(x)` changes its sign in `(-oo,oo)` is___

Explanation

To determine the number of times \( f'(x) \) changes its sign in the interval \( (-\infty, \infty) \), we will follow these steps:

### Step 1: Define the function piecewise
The function \( f(x) \) is given as:
\[ f(x) = \begin{cases} x + 2 & \text{for } x < -1 \\ x^2 & \text{for } -1 \leq x < 1 \\ (x - 2)^2 & \text{for } x \geq 1 \end{cases} \]

### Step 2: Differentiate the function
Now, we will differentiate \( f(x) \) in each piece of the piecewise function:
- For \( x < -1 \):
\[ f'(x) = 1 \]
- For \( -1 \leq x < 1 \):
\[ f'(x) = 2x \]
- For \( x \geq 1 \):
\[ f'(x) = 2(x - 2) = 2x - 4 \]

### Step 3: Identify critical points
Next, we need to find the points where \( f'(x) = 0 \) or is undefined:
- For \( f'(x) = 1 \) (for \( x < -1 \)), it does not change sign.
- For \( f'(x) = 2x \) (for \( -1 \leq x < 1 \)):
\[ 2x = 0 \implies x = 0 \]
- For \( f'(x) = 2x - 4 \) (for \( x \geq 1 \)):
\[ 2x - 4 = 0 \implies x = 2 \]

### Step 4: Analyze sign changes
Now we will analyze the sign of \( f'(x) \) in the intervals determined by the critical points:
1. For \( x < -1 \):
- \( f'(x) = 1 \) (positive)
2. For \( -1 \leq x < 0 \):
- \( f'(x) = 2x \) (negative when \( x < 0 \))
3. For \( 0 < x < 1 \):
- \( f'(x) = 2x \) (positive)
4. For \( 1 \leq x < 2 \):
- \( f'(x) = 2x - 4 \) (negative when \( x < 2 \))
5. For \( x \geq 2 \):
- \( f'(x) = 2x - 4 \) (positive)

### Step 5: Count the sign changes
Now we summarize the sign changes:
- From positive (1) to negative (at \( x = 0 \)) - **1 change**
- From negative (at \( x = 0 \)) to positive (at \( x = 1 \)) - **1 change**
- From positive (at \( x = 1 \)) to negative (at \( x = 2 \)) - **1 change**
- From negative (at \( x = 2 \)) to positive (for \( x > 2 \)) - **1 change**

### Conclusion
Thus, the total number of times \( f'(x) \) changes its sign in the interval \( (-\infty, \infty) \) is:
\[ \text{Total changes} = 4 \]

### Final Answer
The number of times \( f'(x) \) changes its sign in \( (-\infty, \infty) \) is **4**.

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