Local max and min at a if lim f(x) approaching a does not exist

• merry
In summary, the conversation was about the definition of local extrema and whether or not the function needs to exist on both sides of the point to have a local max or min. The question was clarified to be a quibble over definition, with the general consensus being that it is better to consider only interior points for pedagogical reasons. However, one person's university professor stated that local extrema can exist at end points, contradicting what was taught in high school.
merry
Hi,
I understand that the local max of a function is the point at which the y value of the function is larger than the neighbouring y values of the function.
Say we're considering the local max at a of a function f(x). Does the function have to exist on both sides of a for (a,f(a)) to be a local max? (consider the same situation for a local min).
In short, if the fuction exists at [a, b] $$\epsilon$$$$\Re$$ only,
can there be a local max or min at (a, f(a)) or (b, f(b)) ?
My high school professor said that to have local extrema, the function should exist on either side of the point. However, I believe my University prof said that this is not the case.
Could someone please clarify as to which one is the case?
Thanks a ton!
Merry

It appears to be a quibble over definition. If you have a function defined over a finite interval, the end point values will almost always be local extrema (unless the function is constant near the end point). For pedogogical reasons, it is better to consider only interior points.

mathman said:
It appears to be a quibble over definition. If you have a function defined over a finite interval, the end point values will almost always be local extrema (unless the function is constant near the end point). For pedogogical reasons, it is better to consider only interior points.

Thanks! =D
I asked my Uni prof and he said that our high school textbooks are wrong and that local extrema can exist at end points xD

1. What does it mean for a function to have a local maximum or minimum at a point where the limit does not exist?

Having a local maximum or minimum at a point a where the limit of the function f(x) as x approaches a does not exist means that the function either has a sharp peak or valley at that point, or there is a discontinuity in the function at that point. In other words, the function does not approach a specific value as x gets closer to a, but instead has a sudden change in behavior at that point.

2. How can we determine if a point a is a local maximum or minimum when the limit of the function does not exist?

To determine if a point a is a local maximum or minimum when the limit of the function does not exist, we can use the first derivative test. This involves finding the first derivative of the function and evaluating it at the point a. If the derivative is positive at a, then a is a local minimum. If the derivative is negative at a, then a is a local maximum.

3. Can a function have a local maximum or minimum at a point where the limit exists?

Yes, it is possible for a function to have a local maximum or minimum at a point where the limit of the function exists. In this case, the function would approach a specific value as x gets closer to the point a, and the first derivative test would not be applicable. However, the function could still have a sharp peak or valley at that point, making it a local maximum or minimum.

4. How can we graph a function that has a local maximum or minimum at a point where the limit does not exist?

To graph a function that has a local maximum or minimum at a point a where the limit does not exist, we can plot the point a and then use the behavior of the function on either side of a to determine the shape of the graph. If the function approaches a sharp peak or valley at a, the graph would have a sharp point at a. If the function has a discontinuity at a, the graph would have a gap or hole at a.

5. Is it possible for a function to have multiple local maximum or minimum points where the limit does not exist?

Yes, it is possible for a function to have multiple local maximum or minimum points where the limit of the function as x approaches those points does not exist. This could happen if the function has multiple sharp peaks or valleys, or if there are multiple discontinuities in the function. In this case, the first derivative test would need to be applied at each point to determine if it is a local maximum or minimum.

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