Answer:

To determine whether the function has a critical point at x = 2, we need to find the derivative of f(x) and check if it equals zero at x = 2.

Derivative of f(x):
f'(x) = 17(x-2)^16 * (x-5)^24 + 24(x-2)^17 * (x-5)^23

Evaluating f'(2):
f'(2) = 17(2-2)^16 * (2-5)^24 + 24(2-2)^17 * (2-5)^23
= 0 + 0
= 0

Since the derivative f'(x) equals zero at x = 2, we have a critical point at x = 2.

Now, let's analyze the behavior of the function around x = 2 to determine whether it has a maximum, minimum, or neither at this point.

Behavior of f(x) near x = 2:

To determine the behavior of f(x) near x = 2, we can examine the signs of the factors (x-2) and (x-5).

When x < 2,="" both="" factors="" will="" be="" />
(x-2) < 0="" and="" (x-5)="" />< />

When 2 < x="" />< 5,="" (x-2)="" will="" be="" positive="" and="" (x-5)="" will="" be="" />
(x-2) > 0 and (x-5) < />

When x > 5, both factors will be positive:
(x-2) > 0 and (x-5) > 0

Conclusion:

Based on the behavior of the function near x = 2, we can conclude that:

- The function has a critical point at x = 2.
- When x < 2,="" the="" function="" is="" />
- When 2 < x="" />< 5,="" the="" function="" is="" />
- When x > 5, the function is positive.

Therefore, the function does not have a maximum or minimum at x = 2. It simply changes sign from negative to positive as x passes through 2.