Understood. Here is the detailed explanation of the answer:

Given Function:
The given function is f(x) = max(5x, 52 - 2x^2), where x is any positive real number.

Finding the Minimum Possible Value of f(x):
To find the minimum possible value of f(x), we need to analyze the two cases:
1. When 5x is the maximum value.
2. When 52 - 2x^2 is the maximum value.

Case 1: When 5x is the Maximum Value
When 5x is the maximum value, it means that 5x is greater than or equal to 52 - 2x^2. We can set up the following inequality:
5x ≥ 52 - 2x^2

Solving the Inequality:
To solve the inequality, we can rearrange it:
2x^2 + 5x - 52 ≤ 0

We can factorize the quadratic equation:
(2x - 13)(x + 4) ≤ 0

The roots of the equation are x = 13/2 and x = -4.
Since x is a positive real number, we can discard the negative root.
So, the range of x is x ≤ 13/2.

Case 2: When 52 - 2x^2 is the Maximum Value
When 52 - 2x^2 is the maximum value, it means that 52 - 2x^2 is greater than or equal to 5x. We can set up the following inequality:
52 - 2x^2 ≥ 5x

Solving the Inequality:
To solve the inequality, we can rearrange it:
2x^2 + 5x - 52 ≥ 0

We can factorize the quadratic equation:
(2x - 13)(x + 4) ≥ 0

The roots of the equation are x = 13/2 and x = -4.
Since x is a positive real number, we can discard the negative root.
So, the range of x is x ≥ 13/2.

Combining the Ranges:
From both cases, we have the range of x as x ≤ 13/2 and x ≥ 13/2.
Since the range is restricted to x = 13/2, we can conclude that the minimum possible value of f(x) is achieved when x = 13/2.

Calculating f(x) at x = 13/2:
Substituting x = 13/2 into the given function,
f(13/2) = max(5 * 13/2, 52 - 2 * (13/2)^2)
= max(65/2, 52 - 2 * (169/4))
= max(65/2, 52 - 338/4)
= max(65/2, 52 - 169/2)
= max(65/2, -117/2)
= 65/2

Therefore, the minimum possible value of f(x) is 65/2, which is equal to 32.5.

Conclusion:
The correct answer is