Finding the local minimum of a graph

In summary, the conversation discusses the definition of a local minimum and how it applies to a given graph. The question asks for the local minimums of the graph and there is a discrepancy between the provided answer key and the suggested answers. The conversation delves into why certain points are or are not considered local minimums and clarifies that the function does not need to be continuous to have a local minimum. Ultimately, the conversation determines that x=1 is indeed a local minimum and x=6 is a local maximum.
  • #1
JustinLiang
81
0

Homework Statement


The question provides a graph and asks for the local minimums. I attached a picture with the graph.

2. The attempt at a solution
I said the local minima are when x=0,2,5.

However the answer key suggests they are at 1,2,5.
Could someone please explain why 1 is a local minimum? It is just a point on the graph...
Maybe the answer key is wrong.
 

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  • #2
Recall how "local minimum" is defined.

ehild
 
  • #3
ehild said:
Recall how "local minimum" is defined.

ehild

So local minimum tells us that a y value near the point is always greater than the y value of the point. That is why we can call it the local minimum. However, in this case the dot is no continuous, so there is no y value near it to compare?

I understand why 0 cannot be a local minimum but I fail see see why 1 can be.
 
  • #4
Are the y(x) values near to x=1 higher than y(1)? The function does not need to be continuous to have a local minimum. ehild
 
  • #5
ehild said:
Are the y(x) values near to x=1 higher than y(1)? The function does not need to be continuous to have a local minimum.


ehild

Ah okay, but what about x=6? Values near it are both lower, isn't that a local minimum then?
 
  • #6
If all values near x=6 are lower than y(6) so y(6) is higher then anything else nearby, is it a minimum?

ehild
 
  • #7
JustinLiang said:
Ah okay, but what about x=6? Values near it are both lower, isn't that a local minimum then?
No, its a local maximum!
 
  • #8
HallsofIvi,

I know that you know it, (and I hope you think that I also know it) but I asked the OP...:wink:


ehild
 

FAQ: Finding the local minimum of a graph

What is a local minimum in a graph?

A local minimum in a graph is a point where the curve reaches its lowest value within a specific interval, but it may not be the absolute lowest point on the entire curve.

Why is finding the local minimum important?

Finding the local minimum is important because it can help identify the optimal value or solution in a dataset. In optimization problems, the local minimum represents the best possible solution within a given range, and finding it can help improve efficiency and accuracy in decision making.

What methods can be used to find the local minimum of a graph?

There are several methods that can be used to find the local minimum of a graph, such as the gradient descent method, the Newton-Raphson method, and the golden section search. These methods use different approaches to iteratively refine the estimate of the local minimum until a satisfactory solution is reached.

Can a graph have more than one local minimum?

Yes, a graph can have more than one local minimum. This is often the case in complex datasets with multiple variables and constraints. In such cases, it is important to carefully analyze the data and use appropriate methods to accurately locate all the local minima.

Are there any limitations to finding the local minimum of a graph?

Yes, there are limitations to finding the local minimum of a graph. In some cases, the local minimum may not be the optimal solution, and therefore, it is important to carefully consider the data and the problem at hand before making any decisions based on the local minimum.

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