F(x)=xabs(x) show f '(0) exists

Thread starter stl11
Start date Oct 18, 2011

In summary, the function f(x) = x| x | represents a piecewise function that takes the absolute value of x and multiplies it by x, creating a "V" shape with a point at (0,0) and slopes of 1 and -1 on the left and right branches. To determine if the derivative of f(x) exists at x = 0, we need to take the limit of the function as x approaches 0 from both sides and if the limits are equal, the derivative exists and is equal to 0. The formula for finding the derivative of f(x) = x| x | is f'(x) = 2x if x > 0 and f'(x) = -2

Oct 18, 2011

stl11

f(x)=xabs(x) show f '(0) exists

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Oct 18, 2011

daveb

What have you tried so far. You'll have to show some work first showing you attempted it before we can help you.

1. What does the function f(x) = x| x | represent?

The function f(x) = x| x | represents a piecewise function that takes the absolute value of x and multiplies it by x. This means that the function will have a "V" shape with the point at (0,0) and the left and right branches having a slope of 1 and -1 respectively.

2. How do you determine if the derivative of f(x) exists at x = 0?

To determine if the derivative of f(x) exists at x = 0, we need to take the limit of the function as x approaches 0 from both the left and right sides. If the left and right limits are equal, then the derivative exists at x = 0. In this case, the derivative of f(x) at x = 0 is 0.

3. What is the formula for finding the derivative of f(x) = x| x |?

The formula for finding the derivative of f(x) = x| x | is f'(x) = 2x if x > 0 and f'(x) = -2x if x < 0. This means that the derivative is a piecewise function with a slope of 2 on the left side of the "V" and a slope of -2 on the right side of the "V". At x = 0, the derivative is 0.

4. Is the function f(x) = x| x | differentiable at x = 0?

Yes, the function f(x) = x| x | is differentiable at x = 0. As mentioned in the previous questions, the left and right limits of the function at x = 0 are equal, meaning that the derivative exists at this point and is equal to 0.

5. Can the function f(x) = x| x | be extended to become a continuous function?

Yes, the function f(x) = x| x | can be extended to become a continuous function by assigning a value of 0 at x = 0. This means that the function will have a "V" shape with a point at (0,0) and a connected line on either side with a slope of 1 and -1 respectively. This extended function will be continuous at all points, including x = 0.