Finding the local minimum of a graph

[JustinLiang]

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.

 

[ehild]

Recall how "local minimum" is defined.

ehild

 

[JustinLiang]

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.

 

[ehild]

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

 

[JustinLiang]

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?

 

[ehild]

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

 

[HallsofIvy]

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!

 

[ehild]

HallsofIvi,

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


ehild