A problem about differentiability

In summary, the conversation is about a question on differentiability and the speaker has found a solution to it. They ask the listener to check it and mention that the limit value of functions equals zero when x goes infinity. The listener has some trouble understanding the first picture, but the main problem is determining if certain conditions are met, then there is no twice differentiable continuation of the function on the real numbers. The continuity of the second derivative is important for the proof.
  • #1
rasi
18
0
as you know i have been asked a question which no no way i couldn't tackle it. and its is about differentiabilty. at long last i found a solution. i want to share with you. could you check out please. thanks for now.this is the question.
5625-1.jpg

and this is my solution.(i assume that when x goes infinity, the limit value of functions equals zero.)
part1
1.jpg

part2
2.jpg
 
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  • #2
I cannot decode the first picture. However, there are functions which locally violate the assertion. So the problem comes down to: If ##f(x_0)^2 \leq 1## and ##f'(x_0)^2 +f''(x_0)\leq 1 ## and ##f(x_0)^2 +f'(x_0)^2> 1##, then there is no twice differentiable continuation of ##f## on ##\mathbb{R}##. This means on the other hand, that the continuity of ##f''(x)## is crucial for the proof!
 

Related to A problem about differentiability

1. What is differentiability?

Differentiability is a mathematical concept that refers to the smoothness of a function. A function is differentiable if it has a well-defined derivative at every point in its domain. This means that the function has a unique tangent line at each point, and small changes in the input result in small changes in the output.

2. How do you determine if a function is differentiable?

To determine if a function is differentiable, you can use the definition of differentiability, which states that a function is differentiable if the limit of the difference quotient exists at every point in its domain. Alternatively, you can check for continuity and the existence of a derivative using the rules of differentiation for different types of functions.

3. What is the difference between differentiability and continuity?

Continuity and differentiability are related concepts, but they are not the same. Continuity refers to the smoothness of a function at a point, while differentiability refers to the smoothness of a function over an entire interval. A function can be continuous but not differentiable, but a function cannot be differentiable without being continuous.

4. Can a function be differentiable at a point but not on an entire interval?

Yes, a function can be differentiable at a point but not on an entire interval. In this case, the function would have a well-defined derivative at the specific point, but it would not be differentiable over the entire interval due to the existence of a vertical tangent or a discontinuity.

5. What is the importance of differentiability in mathematics and science?

Differentiability plays a crucial role in many areas of mathematics and science. It is used in calculus to solve optimization problems, in physics to calculate rates of change and motion, and in economics to model and predict changes in quantities. It is also a fundamental concept in the study of smooth functions and their properties.

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