If the function `f(x)=| x^2 + a | x | + b |` has exactly three points of non-derivability, then

Options

A. (a)`b=0, alt0`
B. (b)`a lt0, a in R`
C. (c)`bgt 0, a in R`
D. (d)All of these

Answer

To solve the problem, we need to analyze the function \( f(x) = |x^2 + a| |x| + b| \) and determine the conditions under which it has exactly three points of non-derivability.

### Step 1: Identify the points of non-derivability
The function \( f(x) \) involves absolute values, which can create points of non-derivability where the expression inside the absolute value is zero. We need to find the points where:

1. \( x^2 + a = 0 \)
2. \( |x| = 0 \) (which occurs at \( x = 0 \))

### Step 2: Solve for the roots of \( x^2 + a = 0 \)
The equation \( x^2 + a = 0 \) can be rewritten as:

\[ x^2 = -a \]

For this equation to have real roots, \( a \) must be less than or equal to zero. The roots will be:

\[ x = \pm \sqrt{-a} \]

### Step 3: Count the points of non-derivability
The points of non-derivability are:

1. \( x = 0 \) (from \( |x| \))
2. \( x = \sqrt{-a} \) (from \( x^2 + a = 0 \))
3. \( x = -\sqrt{-a} \) (from \( x^2 + a = 0 \))

### Step 4: Determine conditions for exactly three points
To have exactly three points of non-derivability, one of the roots from \( x^2 + a = 0 \) must coincide with \( x = 0 \). This occurs when:

\[ \sqrt{-a} = 0 \Rightarrow -a = 0 \Rightarrow a = 0 \]

In this case, the function simplifies to:

\[ f(x) = |x^2| |x| + b = |x|^3 + b \]

### Step 5: Analyze the value of \( b \)
Since we want exactly three points of non-derivability, we need to ensure that \( b \) does not introduce any additional points of non-derivability. The function \( |x|^3 + b \) is differentiable everywhere except at \( x = 0 \), and thus we have:

1. \( x = 0 \) (from \( |x| \))
2. Two additional points from the roots of \( x^2 + 0 = 0 \) which are \( x = 0 \) (double root).

Thus, we conclude that:

- \( a = 0 \)
- \( b \) can be any real number.

### Conclusion
The conditions for the function \( f(x) = |x^2 + a| |x| + b| \) to have exactly three points of non-derivability are:

- \( a = 0 \)
- \( b \) can be any real number.

### Final Answer
The correct option is \( a = 0 \) and \( b \) can be any real number.