If three positive real numbers a,b,c, are in A.P. such that `a*b*c=4` , then the minimum value of b is :

Options

A. a)`2^(1//2)`
B. b)`2^(1//3)`
C. c)`2^(2//3)`
D. d)`2^(3//2)`

Answer

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

### Step 1: Define the terms in A.P.
Since \( a, b, c \) are in A.P., we can express them in terms of \( b \) and a common difference \( d \):
- Let \( a = b - d \)
- Let \( b = b \)
- Let \( c = b + d \)

### Step 2: Write the product of \( a, b, c \)
Now, we can express the product \( a \cdot b \cdot c \):
\[ a \cdot b \cdot c = (b - d) \cdot b \cdot (b + d) \]
This can be simplified to:
\[ = b(b^2 - d^2) = b^3 - b \cdot d^2 \]

### Step 3: Set up the equation
We know from the problem statement that:
\[ b^3 - b \cdot d^2 = 4 \]
Rearranging gives:
\[ b^3 = 4 + b \cdot d^2 \]

### Step 4: Minimize \( b \)
To minimize \( b \), we can use the method of Lagrange multipliers or simply analyze the equation. We can express \( d^2 \) in terms of \( b \):
\[ d^2 = \frac{b^3 - 4}{b} \]
Since \( d^2 \) must be non-negative (because \( d \) is a real number), we have:
\[ b^3 - 4 \geq 0 \implies b^3 \geq 4 \implies b \geq 4^{1/3} \]

### Step 5: Calculate \( 4^{1/3} \)
Calculating \( 4^{1/3} \):
\[ 4^{1/3} = (2^2)^{1/3} = 2^{2/3} \]

### Conclusion
Thus, the minimum value of \( b \) is:
\[ b_{\text{min}} = 2^{2/3} \]

### Final Answer
The minimum value of \( b \) is \( 2^{2/3} \), which corresponds to option (c).

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