differentiable once but not twice? (ft. Oreo)

TL;DR

This video discusses the possibility of having a function that is differentiable at a specific point but its second derivative does not exist.

Transcript

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Key Insights

• 🎮 The video showcases a whiteboard tool for explaining mathematical concepts.
• 😥 It is possible for a function to have a first derivative at a specific point but not have a second derivative at that point.
• ❓ The absolute value function is a well-known example that satisfies this condition.
• ❓ By manipulating the function's derivative, it is possible to create a function that meets the given condition.
• 😥 The graph of such a function may exhibit a jump or change in slope at the point in question.
• ❓ This concept can be extended to other functions by adjusting their definitions or derivatives.

Questions & Answers

Q: Can a function be differentiable at a point, but its second derivative does not exist?

Yes, it is possible for a function to have a first derivative at a certain point while the second derivative does not exist at that point. This can be demonstrated using the absolute value function as an example.

Q: How can a function be constructed to satisfy this condition?

To create a function that is differentiable at a point but has a non-existent second derivative, you can use the idea of making the derivative of the function equal to the absolute value of x. This can be achieved by defining the function piecewise and adjusting the exponents accordingly.

Q: What happens to the graph of the function in this scenario?

When a function satisfies the condition of being differentiable at a specific point but having a non-existent second derivative, the graph may exhibit a sharp change or jump in its slope at that point. This can be observed in the given example with the absolute value function.

Q: Can this concept be extended to other functions?

Yes, the concept of having a function that is differentiable at a certain point but has a non-existent second derivative can be extended to other functions as well. The key is to manipulate the function and its derivatives to meet the desired condition.

Summary & Key Takeaways

• The video showcases a whiteboard tool that will be used to explain the concept of differentiability and second derivatives.

• The question posed is whether it is possible for a function to have a first derivative at a certain point, but the second derivative does not exist at that point.

• The video provides an example using the absolute value function and demonstrates how to create a function that satisfies the given condition. The graph of the function is also shown.