If three positive real numbers `a, b, c` are in `AP` with `abc = 64,` then minimum value of `b` is:

Options

A. 4
B. 3
C. 2
D. `1/2`

Answer

To find the minimum value of \( b \) given that three positive real numbers \( a, b, c \) are in arithmetic progression (AP) and their product \( abc = 64 \), we can follow these steps:

### Step 1: Understand the condition of AP
Since \( a, b, c \) are in AP, we can express the relationship between them:
\[ 2b = a + c \quad \text{(1)} \]

### Step 2: Use the product condition
We know that:
\[ abc = 64 \quad \text{(2)} \]

### Step 3: Express \( a \) and \( c \) in terms of \( b \)
From equation (1), we can express \( a \) and \( c \) in terms of \( b \):
Let \( a = b - d \) and \( c = b + d \) for some positive \( d \). Then:
\[ abc = (b - d)b(b + d) = b(b^2 - d^2) = 64 \quad \text{(3)} \]

### Step 4: Find the minimum value of \( b \)
To minimize \( b \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. According to AM-GM:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
Substituting \( abc = 64 \):
\[ \frac{a + b + c}{3} \geq \sqrt[3]{64} = 4 \]
Thus:
\[ a + b + c \geq 12 \quad \text{(4)} \]

### Step 5: Substitute \( a + c \) in terms of \( b \)
From equation (1), we know:
\[ a + c = 2b \]
Substituting this into (4):
\[ 2b + b \geq 12 \]
This simplifies to:
\[ 3b \geq 12 \]
Dividing both sides by 3:
\[ b \geq 4 \]

### Conclusion
Thus, the minimum value of \( b \) is:
\[ \boxed{4} \]