F(x)= 2|x| |x 2|- ||x 2|-2|x|| has local minimum or local maximum at...
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.
F(x)= 2|x| |x 2|- ||x 2|-2|x|| has local minimum or local maximum at...
Local Minimum and Maximum of F(x) = 2|x| - |x^2| - ||x^2| - 2|x||
To determine whether the function F(x) = 2|x| - |x^2| - ||x^2| - 2|x|| has a local minimum or maximum at a particular point x, we need to analyze the behavior of the function around that point.
Step 1: Find the critical points
The critical points of a function occur where the derivative is either zero or undefined. Let's find the derivative of F(x) to locate the critical points:
F'(x) = 2 * sign(x) - 2x * sign(x^2) - 2 * sign(|x^2| - 2|x|) * (2x * sign(x) - 2 * sign(x))
Step 2: Analyze the critical points
To determine whether a critical point is a local minimum or maximum, we need to evaluate the second derivative at that point. Let's find the second derivative of F(x):
F''(x) = 2 * delta(x) - 2 * delta(x^2) - 2 * delta(|x^2| - 2|x|) * (2x * delta(x) - 2 * delta(x))
Step 3: Identify the local minimum and maximum
To identify the local minimum and maximum, we need to analyze the sign changes of the first and second derivatives around the critical points.
Case 1: F''(x) > 0
If the second derivative F''(x) is positive, it indicates that the function is concave up, which means it has a local minimum at that point.
Case 2: F''(x) < />
If the second derivative F''(x) is negative, it indicates that the function is concave down, which means it has a local maximum at that point.
Case 3: F''(x) = 0
If the second derivative F''(x) is zero, we cannot determine the concavity of the function at that point. Additional analysis is required, such as evaluating the first derivative or the function itself.
Using the above steps, we can analyze the critical points to determine whether the function F(x) = 2|x| - |x^2| - ||x^2| - 2|x|| has a local minimum or maximum at a particular point x.