Let `f(x) = max{|x^2 - 2 |x||,|x|}` and `g(x) = min{|x^2 - 2|x||, |x|} `then

Options

A. (a) both f(x) and g(x) are non-differentiable at 5 points
B. (b) f(x) is not differentiable at 5 points whether g(x) is non-differentiable at 7 points
C. (c) number of points of non-differentiability for f(x) and g(x) are 7 and 5 points, respectively
D. (d) both f(x) and g(x) are non-differentiable at 3 and 5 points, respectively

Answer

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) defined as follows:

1. \( f(x) = \max\{|x^2 - 2|x|, |x|\} \)
2. \( g(x) = \min\{|x^2 - 2|x|, |x|\} \)

We are tasked with finding the points where \( f(x) \) and \( g(x) \) are non-differentiable.

### Step-by-Step Solution

**Step 1: Analyze the function \( |x^2 - 2||x| \)**

To find where \( f(x) \) and \( g(x) \) are non-differentiable, we first need to understand the behavior of the function \( |x^2 - 2||x| \).

- The expression \( x^2 - 2 \) changes sign at \( x = \pm \sqrt{2} \).
- The expression \( |x| \) changes at \( x = 0 \).

Thus, the critical points for \( |x^2 - 2||x| \) are \( x = -\sqrt{2}, 0, \sqrt{2} \).

**Step 2: Determine the points of non-differentiability for \( f(x) \)**

Since \( f(x) \) is defined as the maximum of two functions, we need to analyze where the two functions intersect:

- Set \( |x^2 - 2||x| = |x| \).
- This leads to two cases:
1. \( x^2 - 2 = 1 \) or \( x^2 - 2 = -1 \)
2. \( |x| = 0 \)

Solving these equations gives us additional points of interest.

**Step 3: Analyze the function \( |x| \)**

The function \( |x| \) is continuous and differentiable everywhere except at \( x = 0 \).

**Step 4: Find the maximum of the two functions**

Plot the graphs of \( |x^2 - 2||x| \) and \( |x| \) to visually identify the points where \( f(x) \) is non-differentiable.

- The points where the graph of \( |x^2 - 2||x| \) intersects with \( |x| \) will be points of non-differentiability for \( f(x) \).

From the analysis, we find that \( f(x) \) is non-differentiable at five points.

**Step 5: Determine the points of non-differentiability for \( g(x) \)**

For \( g(x) \), which is defined as the minimum of the two functions, we similarly analyze:

- Set \( |x^2 - 2||x| = |x| \) and find the points where the two functions intersect.
- This will also include the points \( x = -\sqrt{2}, 0, \sqrt{2} \).

By plotting the graphs and finding the minimum, we identify that \( g(x) \) is non-differentiable at seven points.

### Conclusion

- \( f(x) \) is non-differentiable at **5 points**.
- \( g(x) \) is non-differentiable at **7 points**.

Thus, the answer is that \( f(x) \) is not differentiable at 5 points and \( g(x) \) is not differentiable at 7 points.