Solution:

To find the minimum value of the given function, we need to find the critical points of the function.

Critical points are the points where the derivative of the function is zero or the derivative does not exist.

Let's find the derivative of the given function.

f(x) = |x - 2| |2.5 - x| |3.6 - x|

f'(x) = [(x - 2) / |x - 2|] |2.5 - x| |3.6 - x| + |x - 2| [(2.5 - x) / |2.5 - x|] |3.6 - x| + |x - 2| |2.5 - x| [(x - 3.6) / |3.6 - x|]

Now, to find the critical points, we need to set f'(x) = 0 and solve for x.

[(x - 2) / |x - 2|] |2.5 - x| |3.6 - x| + |x - 2| [(2.5 - x) / |2.5 - x|] |3.6 - x| + |x - 2| |2.5 - x| [(x - 3.6) / |3.6 - x|] = 0

We can simplify this equation by dividing both sides by the common factor |x - 2| |2.5 - x| |3.6 - x|.

[(x - 2) / |x - 2|] + [(2.5 - x) / |2.5 - x|] + [(x - 3.6) / |3.6 - x|] = 0

Now, we can solve this equation by considering the three cases:

Case 1: x < />

In this case, we have:

[(x - 2) / (2 - x)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (x - 3.6)] = 0

Simplifying this equation, we get:

x = 2.4

Case 2: 2.5 < x="" />< />

In this case, we have:

[(x - 2) / (x - 2)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (x - 3.6)] = 0

This equation is not defined at x = 2.5 and x = 3.6. However, we can see that the function is continuous in this interval, and it attains its minimum value at x = 2.5. Therefore, x = 2.5 is a critical point.

Case 3: x > 3.6

In this case, we have:

[(x - 2) / (x - 2)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (3.6 - x)] = 0

Simplifying this equation, we get:

x = 3

Therefore, the critical points of the function are x