Minimum value of ` |z_1 + 1 | + |z_2 + 1 | + |z _1 z _2 + 1 | if [ z_1 | = 1 and |z_2 | = 1 ` is ________.

Explanation

To find the minimum value of \( |z_1 + 1| + |z_2 + 1| + |z_1 z_2 + 1| \) given that \( |z_1| = 1 \) and \( |z_2| = 1 \), we can follow these steps:

### Step 1: Understand the Modulus Condition
Since \( |z_1| = 1 \) and \( |z_2| = 1 \), both \( z_1 \) and \( z_2 \) lie on the unit circle in the complex plane. Therefore, we can express them as:
\[ z_1 = e^{i\theta_1} \quad \text{and} \quad z_2 = e^{i\theta_2} \]
for some angles \( \theta_1 \) and \( \theta_2 \).

### Step 2: Rewrite the Expression
We need to evaluate:
\[ |z_1 + 1| + |z_2 + 1| + |z_1 z_2 + 1| \]

### Step 3: Analyze Each Term
1. **For \( |z_1 + 1| \)**:
\[ |z_1 + 1| = |e^{i\theta_1} + 1| = \sqrt{(1 + \cos \theta_1)^2 + \sin^2 \theta_1} = \sqrt{2 + 2\cos \theta_1} = \sqrt{2(1 + \cos \theta_1)} = 2 \cos\left(\frac{\theta_1}{2}\right) \]
(using the identity \( 1 + \cos \theta = 2 \cos^2(\theta/2) \)).

2. **For \( |z_2 + 1| \)**:
Similarly,
\[ |z_2 + 1| = 2 \cos\left(\frac{\theta_2}{2}\right) \]

3. **For \( |z_1 z_2 + 1| \)**:
\[ |z_1 z_2 + 1| = |e^{i(\theta_1 + \theta_2)} + 1| = 2 \cos\left(\frac{\theta_1 + \theta_2}{2}\right) \]

### Step 4: Combine the Terms
Now we can rewrite the entire expression:
\[ |z_1 + 1| + |z_2 + 1| + |z_1 z_2 + 1| = 2 \cos\left(\frac{\theta_1}{2}\right) + 2 \cos\left(\frac{\theta_2}{2}\right) + 2 \cos\left(\frac{\theta_1 + \theta_2}{2}\right) \]

### Step 5: Find the Minimum Value
To minimize the expression, we can use the fact that the cosine function achieves its maximum value of 1 when its argument is 0. Thus, the minimum value occurs when:
\[ \theta_1 = 0, \quad \theta_2 = 0 \quad \text{(or any multiples of \( 2\pi \))} \]
This gives:
\[ |z_1 + 1| = 2, \quad |z_2 + 1| = 2, \quad |z_1 z_2 + 1| = 2 \]
So:
\[ |z_1 + 1| + |z_2 + 1| + |z_1 z_2 + 1| = 2 + 2 + 2 = 6 \]

However, we need to find the minimum value of the expression, which occurs when \( z_1 = -1 \) and \( z_2 = -1 \):
\[ |(-1) + 1| + |(-1) + 1| + |(-1)(-1) + 1| = |0| + |0| + |2| = 0 + 0 + 2 = 2 \]

### Final Answer
Thus, the minimum value of \( |z_1 + 1| + |z_2 + 1| + |z_1 z_2 + 1| \) is \(\boxed{2}\).