Explanation:

To find the points at which f(x) attains a local extremum, we need to find the critical points of the function. Critical points occur where the derivative of the function is equal to zero or does not exist.

Step 1: Find the derivative of f(x)
To find the derivative of f(x), we need to consider two cases:
1. x^2 - 25 ≥ 0 (when x^2 - 25 is non-negative)
2. x^2 - 25 < 0="" (when="" x^2="" -="" 25="" is="" />

Case 1: x^2 - 25 ≥ 0
In this case, |x^2 - 25| = x^2 - 25, so the function f(x) becomes f(x) = x^2 - 25.

Taking the derivative of f(x), we get:
f'(x) = 2x

Case 2: x^2 - 25 < />
In this case, |x^2 - 25| = -(x^2 - 25), so the function f(x) becomes f(x) = -(x^2 - 25) = -x^2 + 25.

Taking the derivative of f(x), we get:
f'(x) = -2x

Now, let's find the critical points by setting the derivatives equal to zero and solving for x.

Step 2: Find the critical points
For Case 1, setting f'(x) = 2x = 0, we find x = 0.

For Case 2, setting f'(x) = -2x = 0, we find x = 0.

Therefore, the critical point for both cases is x = 0.

Step 3: Determine the nature of the critical points
To determine whether the critical point is a local minimum or maximum, we need to analyze the second derivative of the function.

Case 1: x^2 - 25 ≥ 0
The second derivative of f(x) = x^2 - 25 is given by:
f''(x) = 2

Since the second derivative is positive (2 > 0), the critical point x = 0 corresponds to a local minimum.

Case 2: x^2 - 25 < />
The second derivative of f(x) = -x^2 + 25 is given by:
f''(x) = -2

Since the second derivative is negative (-2 < 0),="" the="" critical="" point="" x="0" corresponds="" to="" a="" local="" />

Step 4: Find the total number of points where f attains a local extremum
Since the critical point x = 0 corresponds to both a local minimum and a local maximum (depending on the case), there are a total of 2 points where f attains a local extremum.

Therefore, the correct answer is option 'B' (2).