Exact number of zeros for any given continuous function

[Tymick]

I'm in need of sources, articles, mainly anything that can provide information on finding the exact number of zeros for any given continuous function, thanks in advance.

 

[Marin]

I'm in need of sources, articles, mainly anything that can provide information on finding the exact number of zeros for any given continuous function, thanks in advance.

Hi!

What do you mean by 'exact number of zeros'? If that's the intersections of the function with the x-axis all you do is solve the equation: f(x)=0

1. If the function is polynomial read more about factorising and dividing polynomials, the theorem of Vieta and the Horner table (or scheme, or schedule - don't know the English term :( )

2. If the function consists of transcendent and polynomial functions, i.e: f(x)=x^3-sinx+lnx you are unable to do the calculations analytically (in the most cases) - so look at the Newton's approximation method - unfortunately, the number you get will no be 'exact' :(

3. If the function is defined over C - the field of the complex numbers - take a look at the fundamental theorem of algebraThat's all I can think of up to now :)

Best wishes, Marin

[Edit]: If it's up to the number and not the coordinates of the points, try just sketching the graph and counting them. Sketch the graph, using the knowledge from differential calculus (limits, maxima, minima, asymptotes, inflex points)

 

[Tymick]

Thanks marin,

However I already know of those methods, notice I never asked for methods but rather for papers that have been published, or any other source that I could use for research, on the exact number of zeros given any arbitrary interval on any given continuous function, I'm just having some trouble finding papers, as I said above.

(Note: I don't plan to generalize such a thing, the entire concept of it is fairly difficult from where I stand, but it's just for research.)

 

[NoMoreExams]

Go here: http://mathworld.wolfram.com/Root.html. At the bottom are some references you can check out at your local library to see if they will aid you with your research.