Local Minimum and Local Maximum

To determine whether the function F(x) = 2|x| |x^2| - ||x^2| - 2|x|| has a local minimum or local maximum at x, we need to analyze the behavior of the function around that point.

Step 1: Find Critical Points

To find critical points, we need to find the values of x where the derivative of the function is equal to zero or undefined. Let's calculate the derivative of F(x) step by step:

1. Derivative of |x| = x/|x| = x/√(x^2) = x/|x| = sgn(x)
2. Derivative of |x^2| = 2x|x| = 2x * sgn(x)
3. Derivative of ||x^2| - 2|x|| = sgn(|x^2| - 2|x|) * (2|x| - 2sgn(x))

Now, let's simplify the expression for F'(x):

F'(x) = 2|x| * 2x * sgn(x) - sgn(|x^2| - 2|x|) * (2|x| - 2sgn(x))

Step 2: Analyze the Function's Behavior

To determine whether the critical points are local minimums or local maximums, we need to analyze the sign changes of the derivative around those points.

Let's consider the intervals between the critical points and analyze the sign of the derivative within each interval:

Interval (-∞, 0)
- F'(x) = -4x^2 - 2 = -2(2x^2 + 1)
- Since the derivative is negative for all x in this interval, the function is decreasing.

Interval (0, 1)
- F'(x) = 4x^2 - 2 = 2(2x^2 - 1)
- Since the derivative is positive for all x in this interval, the function is increasing.

Interval (1, ∞)
- F'(x) = -4x^2 + 2 = 2(-2x^2 + 1)
- Since the derivative is negative for all x in this interval, the function is decreasing.

Step 3: Conclusion

From the analysis of the derivative, we can see that the function F(x) has no critical points where the derivative is equal to zero. Therefore, it does not have any local minimum or local maximum points.

Summary:
The function F(x) = 2|x| |x^2| - ||x^2| - 2|x|| does not have any local minimum or local maximum points.