- #1
mufq15
- 7
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I need to find an example of such a function. I know that x^2sin(1/x) is differentiable but not C^1, but I'm having trouble extending this to C^2.
A function is considered twice differentiable if its first and second derivatives exist. It is not C^2 if its second derivative is not continuous.
No, a function cannot be C^2 if it is not twice differentiable. C^2 means that a function is twice differentiable and its second derivative is also continuous.
A function that is twice differentiable but not C^2 has a discontinuous second derivative. This means that the function has a sharp change in slope at some point, which can affect its behavior and make it more challenging to analyze.
To determine if a function is twice differentiable but not C^2, you can first find its first and second derivatives. Then, check if the second derivative is continuous at all points. If it is not, then the function is not C^2.
One example is the absolute value function, f(x) = |x|. Its first derivative is f'(x) = 1 for x > 0 and f'(x) = -1 for x < 0. Its second derivative is not continuous at x = 0, as f''(x) does not exist at that point. Another example is the function f(x) = x^(2/3), which has a discontinuous second derivative at x = 0.