Find a line that maximally intersects a given function

In summary, the question is whether a line y=ax+b can intersect an arbitrary function y=f(x) a maximum number of times, and if so, is there a method for finding this line? It is argued that for any function f(x), the function that will intersect it the most times will be g(x)=f(x), as every single point will be intersecting. However, this may not be the case for certain trinomial curves where a proof can be shown. It is suggested that the answer to this question may lie in the study of algebraic geometry and the degree of the curve, as well as the number of multiple points. One method proposed involves finding the roots of a function G(x)=F(x)-Ax-b
  • #1
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Can a line y=ax+b that intersects an arbitrary function y=f(x) a maximum number of times be found in general, where a and b are constants to be determined?
 
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  • #2
Wow, that's a really interesting idea, I have no idea how you would do it, but please let me know if you make any advances in trying to figure this out!
 
  • #3
Do you mean "does such a line exist" or "is there a prescription for finding such a line?"

If the first one, my guess is no. I would construct a function along the lines of sin(x)/x for x>0... alter it slightly to that the bottom of the sin asymptotes to but never touches the line y=0. Then the lines y=const will intersect more and more times as const->0, hence no maximum. I think.
 
  • #4
hmmm good point on that sin function.

I think that perhaps you could possibly argue that for any function f(x), the function that will intersect it the most times will be g(x) where g(x) = f(x). This make sense because every single point will be intersecting. For some reason I think that if you were to come up with some sort of math "recipe" to find the function with the most number of intersections it would yield the function itself, and perhaps in certain cases it would allow other functions to fit the recipe also where the other functions also have infinite intersections, like in Damgo's example. But really I don't know and this is just rambling so take it with a grain of salt...
 
  • #5
damgo
Do you mean "does such a line exist" or "is there a prescription for finding such a line?"
Apparently the latter.

Take a trinomial curve. At most I can intersect that function with a straight line three times. So for certain trinomials one may algebraically show this fact.

(What was your PF 2.0 handle, damgo?)
 
  • #6
I haven't got a proof here, but I'd guess that the answer is yes. You could find a straight line that passed through a function f(x) a maximum number of times. I don't think the line would be unique though, there would probably be a large number of possible lines.

Is there a practical reasoning behind this question? Or is it an interest from the purist side of things (ie. interested in the actual mathematics)?
 
  • #7
It's pretty late at night and I have school tomorrow, so I don't really feel like cultivating this idea, but maybe if you split a continuous function y=f(x) into intervals of x where the function was strictly increasing/decreasing or maybe concave upward/downward and then imposed limits on the a and b of y=ax+b so that the line intersected the function within each interval, it might be possible to obtain a valid range for a and b on a whole so that the function is intersected "maximally". I think this is at least a step in the right direction.
 
  • #8
I believe the general case (a plane intesecting a torus, etc) is an active research area in topology, isn't it?
 
  • #9
oh man am i stupid, you said you wanted a line in y = ax + b form to be maximally intersecting, well I guess that immediately gets rid of that stupid crap i said above, what was i thinking please ignore it.
 
  • #10
very interesting...but...isn't this something that researchers haven't found out...or a conjecture ?
because if the answer existed you would have found it on the internet...
 
  • #11
Pauly Man, pure interest.

bogdan, next to prove that the internet is finite.
 
  • #12
I am pretty ignorant here, but I believe this question is part of the study of (older) algebraic geometry. They ask themselves what is the degree of the curve, how many multiple points there might be, and, yes, what is the maximum number of intersections a generic line might have with the curve. There are strong theorems, and I believe that counting multiple points (tangents and such) you can always find a line that intersects a nondegenerate curve of degree n in n points.
 
  • #13
I can think of perhaps one method.

For a given function F(x) the line
Ax +b intersects it at certain points.

therefore there are x such that

F(x) = Ax + b

hence

F(x) - Ax - b = 0
so we have

G(x) = 0 for G(x) = F(x) - Ax -b

All that is needed now is to find some way of working out the roots of the function G(x). The roots I'm guessing will depend somehow on A and b and so different values of A and b will proboby give different numbers of roots.

Or perhaps they all give the same number of roots but different multiplicities...?

Yes actually.
If F(x) is a polynomial of degree n > 1 then it must have n roots.

Thus G(x) has n roots. Now A and b must be chosen such that as many of the roots as possible are real and so that multiplicities are reduced.

For non-polynomial functions?

It may be possible that this intersection has something to do with the turning points of the graph...

In fact it does. If the graph has an infinite number of turning points, such as the Sine function, then the number of possible intersections a line can make is infinite.

Also, if the function has an infinite number of discontiuities, such as Tan(x), then the number of intersections will be infinite.

So we should restrict F(x) to only those functions with a finite number of turning points and discontinuities.

nes pas?
 
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1. What does it mean to find a line that maximally intersects a given function?

Finding a line that maximally intersects a given function means finding a straight line that crosses the function at the point where the function is at its highest point or peak. This line is known as the tangent line and it represents the maximum rate of change of the function at that point.

2. Why is it useful to find a line that maximally intersects a given function?

By finding the line that maximally intersects a given function, we can determine the maximum value of the function and also the direction in which the function is increasing or decreasing at that point. This information can be useful in optimization problems and in understanding the behavior of the function.

3. What are the steps involved in finding a line that maximally intersects a given function?

The steps involved in finding a line that maximally intersects a given function include:

  1. Calculating the derivative of the function
  2. Setting the derivative equal to zero to find the critical point
  3. Substituting the critical point into the original function to find the corresponding y-value
  4. Using the critical point and the corresponding y-value to find the equation of the tangent line using the point-slope form

4. Can a function have multiple lines that maximally intersect it?

Yes, a function can have multiple lines that maximally intersect it. This can happen if the function has multiple peaks or if the function is not differentiable at the point where it reaches its maximum value.

5. How can we verify if a line maximally intersects a given function?

We can verify if a line maximally intersects a given function by plotting the function and the line on a graph and observing the point where they intersect. If the line is tangent to the function at that point, then it is the maximum intersecting line. Additionally, we can also check if the slope of the line is equal to the derivative of the function at that point.

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