Local Extrema of Functions Exhibiting Self-Similarity

In summary, the speaker has a question about the Weierstrass function and its local extrema. They ask if it is possible for every point of the function to be a local extremum, given its self-similarity. They also question the definition of a neighborhood and whether every point can be considered an extremum. The response is that every point cannot be an extremum, as the definition of extremum involves a neighborhood with more than one point.
  • #1
moonlitdawn
2
0
Hi guys. Today was my last day of Calculus I class, and I’ve had this question simmering in my mind for some time now. Perhaps you guys may enlighten me or point me in the right direction?

Consider the Weierstrass function, which is continuous everywhere but differentiable nowhere. We know that a local extremum, let’s say a local maximum, means that there is a number b, a < b < c, such that f(b) >= f(x) in (a, c). We also know that a local extremum either has a derivative of 0 or is non-differentiable.

My question is: Is it possible for every point of the Weierstrass function to be a local extremum (considering it is a non-constant function), given its self-similarity? If you look at it from one viewing rectangle it appears as if it the function has a maximum/minimum at a point in a given neighborhood, but then you zoom in and zoom in and zoom in. . . .

If not, how does one locate the points in which there is a local extremum?

Sorry if that was a lame question, and thanks in advance!
 
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  • #2
You should recall the definition of extremum. It involves a neighborhood. Can every point be an extremum given this definition?
 
  • #3
Yes, it involves a neighborhood. Well, logically the answer would be no, right? But what counts as a "neighborhood" here?
 
  • #4
Well, use whatever definition you were given in your course of Calculus. Anyway it will have more than one point. So every point cannot be an extremum, because its neighborhood must contain other points, which are non-extremal.
 

1. What is a local extremum?

A local extremum is a point on a function where the function reaches either the highest or lowest value in a specific region, but not necessarily the highest or lowest value overall. It can be thought of as a peak or a valley on a graph.

2. What does it mean for a function to exhibit self-similarity?

A function exhibits self-similarity when it has a repeating pattern at different scales. This means that if you zoom in or out on a portion of the function, it will look similar to the original function.

3. How can you identify local extrema on a graph?

On a graph, local extrema can be identified by looking for points where the slope of the function changes from positive to negative (for a local maximum) or negative to positive (for a local minimum). They can also be identified by looking for points where the tangent line is horizontal.

4. How are local extrema related to self-similarity?

Functions that exhibit self-similarity often have local extrema at each repeating scale. This allows for a more nuanced understanding of the function, as the local extrema can be seen as the "building blocks" of the self-similar pattern.

5. What are some real-world examples of functions exhibiting self-similarity and local extrema?

One example is the Mandelbrot set, a fractal that exhibits self-similarity at different scales. Another example is the coastline of a country, which exhibits self-similarity as you zoom in or out and also has local extrema such as bays and peninsulas.

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