If the positive real numbers a, b and c are in Arithmetic Progression, such that abc = 4, then minimum possible value of b is

Options

A. `2^(3/2)`
B. `2^(2/3)`
C. `2^(1/3)`
D. none of these

Answer

To find the minimum possible value of \( b \) given that \( a, b, c \) are in Arithmetic Progression (AP) and \( abc = 4 \), we can follow these steps:

### Step 1: Express \( a \) and \( c \) in terms of \( b \)
Since \( a, b, c \) are in AP, we can express \( a \) and \( c \) as:
- \( a = b - x \)
- \( c = b + x \)

where \( x \) is the common difference.

### Step 2: Set up the equation for the product
Given that \( abc = 4 \), we substitute \( a \) and \( c \):
\[ (b - x) \cdot b \cdot (b + x) = 4 \]

### Step 3: Simplify the equation
Expanding the left-hand side:
\[ (b - x)(b + x) = b^2 - x^2 \]
Thus, the equation becomes:
\[ b(b^2 - x^2) = 4 \]
or
\[ b^3 - bx^2 = 4 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ bx^2 = b^3 - 4 \]
From this, we can express \( x^2 \):
\[ x^2 = \frac{b^3 - 4}{b} \]

### Step 5: Apply the AM-GM inequality
Since \( a, b, c \) are positive real numbers, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
Substituting \( abc = 4 \):
\[ \frac{(b - x) + b + (b + x)}{3} \geq \sqrt[3]{4} \]
This simplifies to:
\[ \frac{3b}{3} \geq \sqrt[3]{4} \]
Thus,
\[ b \geq \sqrt[3]{4} \]

### Step 6: Find the minimum value of \( b \)
To find the minimum possible value of \( b \), we can calculate:
\[ \sqrt[3]{4} = 2^{2/3} \]

### Step 7: Conclusion
Thus, the minimum possible value of \( b \) is:
\[ b = 2^{2/3} \]