Question about Min and Max for Turning Points and Zeros for Polynomial Functions

[AJ22]

Hi,
I'm having trouble understanding what exactly the min and max represent for polynomial functions by using the degree. The textbook doesn't really clarify this question for Advanced Functions....
For turning points and numbers of zeros, does min and max only mean those values. If you had an odd degree of 3 it has a min of 0 and a max of 2 TURNING POINTS (n-1). Does that exclude the ability for it to have only 1 turning point? And also for zeros, if an odd degree polynomial of 3 means a min of 1 and a max of 3 ZEROS, can it still have just 2 zeros or does it always have to be the set min and max values?

This question doesn't just apply for polynomials with a degree of 3, I just used that as an example. I don't understand if the min and max are set numbers or act as range (between). For zeross does it work as a range and for turning points is it just the minimum and maximum and nothing in between??

 

[lev888]

Hi,
I'm having trouble understanding what exactly the min and max represent for polynomial functions by using the degree. The textbook doesn't really clarify this question for Advanced Functions....
For turning points and numbers of zeros, does min and max only mean those values. If you had an odd degree of 3 it has a min of 0 and a max of 2 TURNING POINTS (n-1). Does that exclude the ability for it to have only 1 turning point? And also for zeros, if an odd degree polynomial of 3 means a min of 1 and a max of 3 ZEROS, can it still have just 2 zeros or does it always have to be the set min and max values?

This question doesn't just apply for polynomials with a degree of 3, I just used that as an example. I don't understand if the min and max are set numbers or act as range (between). For zeross does it work as a range and for turning points is it just the minimum and maximum and nothing in between??

I think what would be very helpful in understanding this is looking at a few graphs. Does the book use graphs in illustrating all those cases?
E.g. take a 3rd degree polynomial. Let's say it looks like this (left to right): it goes up, crosses the x axis, then turns, goes down, turns before crossing the x axis and goes up again. It has one zero. Now take the whole thing and shift it down so that it now crosses the x axis before turning up. Now it has 3 zeros.

 

[AJ22]

Right, so its impossible for it to have 2 just two zeros then? Does the min and max mean that those two numbers are the only options for turning points and for zeros?

 

[Dr.Peterson]

Hi,
I'm having trouble understanding what exactly the min and max represent for polynomial functions by using the degree. The textbook doesn't really clarify this question for Advanced Functions....
For turning points and numbers of zeros, does min and max only mean those values. If you had an odd degree of 3 it has a min of 0 and a max of 2 TURNING POINTS (n-1). Does that exclude the ability for it to have only 1 turning point? And also for zeros, if an odd degree polynomial of 3 means a min of 1 and a max of 3 ZEROS, can it still have just 2 zeros or does it always have to be the set min and max values?

This question doesn't just apply for polynomials with a degree of 3, I just used that as an example. I don't understand if the min and max are set numbers or act as range (between). For zeross does it work as a range and for turning points is it just the minimum and maximum and nothing in between??

I'd like to see exactly how your textbook (or notes) states the theorems or facts you are talking about. That may make it easier to talk about what it means.

But in general, "maximum" means the largest number possible, and "minimum" means the smallest number possible; these terms say nothing about what number is possible in between. So if you are told "this graph has a minimum of 0 and maximum of 2 turning points", then it is merely saying that it has at least 0 and at most 2; it is not saying whether it can have 1. That would be a separate fact to work out by other means. (One way to think about it would be to do what lev888 has suggested.)

Similarly, if you are told that a certain function has a minimum of 1 zero and a maximum of 3, you are not being told whether it can have exactly 2. That may or may not be possible, based on other theorems, not this one.

In each case they are giving you a range, not the only two possibilities.

But in fact, it turns out that in your specific examples, in one case the intermediate number is possible, and in the other it is not! You can figure that out for yourself; but what you have been taught (as far as you have told us) doesn't provide those answers. And for higher degrees, there will be intermediate numbers possible in both cases.

 

[AJ22]

To answer these questions is no problem at all but I don't understand how exactly max and min work..

 

[Dr.Peterson]

I see your statements were not based on theorems, as I thought, but on the kind of thinking lev888 demonstrated. But one thing you didn't mention explicitly is that the possible numbers of turning points for an odd-degree polynomial are all even; do you understand why? This is the reason that not all numbers between the minimum and maximum are possible.

I'd like to see your own answers to these questions, including your reasoning; we can use that reasoning to explain the parts you are unsure of.