To find the minimum value of b, we need to understand the relationship between the numbers a, b, and c in an arithmetic progression (AP) and their product abc. Let's break down the problem into smaller parts.

Understanding Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, a, b, and c are in AP, so we can write them as:

a = b - d
b = b
c = b + d

Here, d represents the common difference between the terms. Since a, b, and c are positive real numbers, d must be positive.

Product of a, b, and c
Given that abc = 64, we can substitute the values of a, b, and c from the AP:

(b - d) * b * (b + d) = 64

Expanding the equation, we get:

b^3 - d^2 * b = 64

Finding the Minimum Value of b
To find the minimum value of b, we need to find the value of d that minimizes the equation. Let's differentiate the equation with respect to d:

d(b^3) / dd - d^2 * db / dd = 0

3b^2 - 2db^2 = 0

Factoring out b^2, we get:

b^2(3 - 2d) = 0

Since b is a positive real number, we can ignore the b^2 term. Therefore, we have:

3 - 2d = 0

Solving for d, we get:

d = 3/2

Substituting d into the AP
Now that we have the value of d, we can substitute it back into the AP to find the values of a, b, and c:

a = b - (3/2)
b = b
c = b + (3/2)

Simplifying these equations, we get:

a = (b - 3/2)
b = b
c = (b + 3/2)

Minimum Value of b
Since the question asks for the minimum value of b, we can substitute the values of a and c into the equation abc = 64:

(b - 3/2) * b * (b + 3/2) = 64

Expanding and simplifying the equation, we get a quadratic equation:

4b^2 - 9 = 64

4b^2 = 73

b^2 = 73/4

Taking the square root, we get:

b = √(73/4)

The value of √(73/4) is approximately 4.028. Since b is a positive real number, the minimum value of b is rounded up to 4.

Therefore, the correct answer is option D) 4.