The number of points at which the function `f(x) = (x-|x|)^(2)(1-x + |x|)^(2)` is not differentiable in the interval `(-3, 4)` is ___

Answer

To determine the number of points at which the function \( f(x) = (x - |x|)^2 (1 - x + |x|)^2 \) is not differentiable in the interval \((-3, 4)\), we will analyze the function based on the definition of the absolute value.

### Step 1: Analyze the function for \( x \geq 0 \)

For \( x \geq 0 \), we have \( |x| = x \). Thus, the function simplifies as follows:

\[ f(x) = (x - x)^2 (1 - x + x)^2 = 0^2 \cdot 1^2 = 0 \]

This means that for \( x \in [0, 4) \), \( f(x) = 0 \), which is a constant function. A constant function is differentiable everywhere in its domain.

### Step 2: Analyze the function for \( x < 0 \)

For \( x < 0 \), we have \( |x| = -x \). Therefore, the function becomes:

\[ f(x) = (x - (-x))^2 (1 - x + (-x))^2 = (x + x)^2 (1 - x - x)^2 = (2x)^2 (1 - 2x)^2 \]

This simplifies to:

\[ f(x) = 4x^2 (1 - 2x)^2 \]

### Step 3: Check differentiability for \( x < 0 \)

The function \( f(x) = 4x^2 (1 - 2x)^2 \) is a polynomial, and polynomial functions are differentiable everywhere. Thus, \( f(x) \) is differentiable for all \( x < 0 \).

### Step 4: Check at the point \( x = 0 \)

Now we need to check the differentiability at the point \( x = 0 \):

- **Right-hand derivative at \( x = 0 \)**:
Since \( f(x) = 0 \) for \( x \geq 0 \), the right-hand derivative is:

\[ f'(0^+) = 0 \]

- **Left-hand derivative at \( x = 0 \)**:
We need to calculate the derivative of \( f(x) \) for \( x < 0 \):

\[ f(x) = 4x^2(1 - 2x)^2 \]

Using the product rule:

\[ f'(x) = 8x(1 - 2x)^2 + 4x^2 \cdot 2(1 - 2x)(-2) = 8x(1 - 2x)^2 - 16x^2(1 - 2x) \]

Now substituting \( x = 0 \):

\[ f'(0^-) = 8(0)(1 - 0)^2 - 16(0^2)(1 - 0) = 0 \]

Since both the right-hand derivative and left-hand derivative at \( x = 0 \) are equal:

\[ f'(0^+) = f'(0^-) = 0 \]

Thus, \( f(x) \) is differentiable at \( x = 0 \).

### Conclusion

Since \( f(x) \) is differentiable for all \( x \) in the interval \((-3, 4)\), we conclude that there are no points in this interval where the function is not differentiable.

The final answer is:

\[ \text{The number of points at which } f(x) \text{ is not differentiable in } (-3, 4) \text{ is } 0. \]

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