If $f(x)=|x^{2}-25|$ for all $x\in R$. The total number of points on R at which f attains a local extremum (minimum or maximum) is:

Question

Accepted Answer

To determine the total number of points on $\mathbb{R}$ at which the function $f(x) = |x^2 - 25|$ attains a local extremum, we need to analyze the behavior of the function.

### Step-by-Step Explanation:

1. **Function Analysis**:
   - The function $f(x) = |x^2 - 25|$ can be rewritten as:
     - $f(x) = x^2 - 25$ when $x^2 \geq 25$
     - $f(x) = 25 - x^2$ when $x^2 < 25$

2. **Critical Points**:
   - To find critical points, we need to consider where the derivative $f'(x)$ is zero or undefined.
   - For $x^2 \geq 25$, $f(x) = x^2 - 25$, so $f'(x) = 2x$.
     - $f'(x) = 0$ gives $x = 0$, but $x = 0$ does not satisfy $x^2 \geq 25$.
   - For $x^2 < 25$, $f(x) = 25 - x^2$, so $f'(x) = -2x$.
     - $f'(x) = 0$ gives $x = 0$, which satisfies $x^2 < 25$.

3. **Boundary Points**:
   - The boundaries between the two cases are at $x = \pm 5$, where $x^2 = 25$.
   - At $x = 5$ and $x = -5$, the function changes from $x^2 - 25$ to $25 - x^2$ or vice versa.

4. **Local Extrema**:
   - At $x = 0$, $f(x) = 25 - x^2$ has a local maximum because the derivative changes sign from positive to negative.
   - At $x = 5$ and $x = -5$, the function transitions between the two cases, and these points are local minima because the function value is zero, which is the lowest possible value for $f(x)$.

5. **Conclusion**:
   - The function $f(x)$ has local extrema at $x = -5$, $x = 0$, and $x = 5$.
   - Therefore, there are **three** points where the function attains a local extremum.

### Explanation of Options:

- **Option 1 (Four)**: Incorrect because there are only three points of local extrema, not four.
- **Option 2 (Three)**: Correct because the function has local extrema at three distinct points: $x = -5$, $x = 0$, and $x = 5$.
- **Option 3 (Two)**: Incorrect because it undercounts the number of local extrema.
- **Option 4 (One)**: Incorrect because it significantly undercounts the number of local extrema.

Thus, the correct answer is Option 2, as the function $f(x) = |x^2 - 25|$ attains local extrema at three points on $\mathbb{R}$.