Understanding the Problem
Given three positive real numbers a, b, and c in Arithmetic Progression (AP) with the condition that abc = 4, we need to find the minimum value of b.
Properties of AP
- Since a, b, and c are in AP, we can express them as:
- a = b - d
- b = b
- c = b + d
- Here, d is the common difference.
Expressing the Product
- The product abc can be rewritten using the expressions of a and c:
- abc = (b - d) * b * (b + d)
- Expanding this, we have:
- abc = b(b^2 - d^2) = b^3 - bd^2
- Setting this equal to 4 gives us the equation:
- b^3 - bd^2 = 4
Applying AM-GM Inequality
- To minimize b, we apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality on a, b, and c:
- (a + b + c) / 3 ≥ (abc)^(1/3)
- Substituting abc = 4:
- (a + b + c) / 3 ≥ (4)^(1/3)
- This implies a + b + c ≥ 3 * (4)^(1/3)
Finding the Minimum Value of b
- Let’s express a + b + c in terms of b:
- a + b + c = (b - d) + b + (b + d) = 3b
- Thus, we have:
- 3b ≥ 3 * (4)^(1/3)
- Simplifying gives b ≥ (4)^(1/3)
Conclusion
- The minimum value of b occurs when a = b = c, hence:
- b = (4)^(1/3) = 4^(1/3) = 2^(2/3)
- Therefore, the minimum value of b is 2^(2/3), which approximately equals 1.587.

(4)1/3 OR (2)2/3.
(Both same and correct)