Explanation:

To find the maximum possible value of f(x), we need to determine the value of x that results in the maximum value of f(x).

Step 1: Finding the critical points
To find the critical points of f(x), we need to determine the values of x where the derivative of f(x) is equal to zero.

Let's find the derivative of f(x) with respect to x:

f'(x) = d/dx(min(2x^2, 52 - 5x))

To find the derivative of min(2x^2, 52 - 5x), we need to consider two cases:

Case 1: When 2x^2 is smaller than 52 - 5x
In this case, the derivative of f(x) with respect to x is equal to the derivative of 2x^2:

f'(x) = d/dx(2x^2) = 4x

Case 2: When 2x^2 is greater than or equal to 52 - 5x
In this case, the derivative of f(x) with respect to x is equal to the derivative of (52 - 5x):

f'(x) = d/dx(52 - 5x) = -5

Step 2: Analyzing the critical points
To find the critical points, we set the derivatives equal to zero and solve for x:

Case 1: When 2x^2 is smaller than 52 - 5x:
4x = 0
x = 0

Case 2: When 2x^2 is greater than or equal to 52 - 5x:
-5 = 0
No solutions exist for this case.

Step 3: Determining the maximum value of f(x)
To determine the maximum value of f(x), we need to evaluate f(x) at the critical points and the endpoints of the domain.

Case 1: Evaluate f(x) at x = 0:
f(0) = min(2(0)^2, 52 - 5(0)) = min(0, 52) = 0

Case 2: Evaluate f(x) at the endpoint of the domain:
Since x is a positive real number, there is no endpoint of the domain.

Conclusion:
Therefore, the maximum possible value of f(x) is 0, which contradicts the given answer of 32. It is possible that there is an error in the given information or question.