Math Is Fun Forum

CIV

Member

Registered: 2014-11-09

Posts: 74

Critical and Inflection points of Rational Functions

I know that with polynomials I solve for the "zeros" of the first and second derivatives and I know I do the same with rational functions but... can someone please clear up where the zeros for each are solved? I was always a bit lost with certain things about rational functions.

Are critical points(extrema) solved in the numerator or both the numerator and denominator? I know that critical points are points where f'(c) = 0 or fails to exist so I'm thinking both the numerator and denominator?

What about possible points of inflection? Numerator or both numerator and denominator? My book doesn't say.

Last edited by CIV (2017-03-06 12:31:36)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Critical and Inflection points of Rational Functions

hi CIV,

When you say 'critical points' do you mean maxima and minima?  I use the term 'turning points' and it means the points where the gradient is zero.  So I find the first derivative and check when it is zero.  That would be the numerator for a rational function.  The second derivative can help to determine which of the three alternatives but this becomes problematic when that is also zero.

Once you have the gradient function you can reliably test its nature by substituting a value of x just left and just right of the point.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

CIV

Member

Registered: 2014-11-09

Posts: 74

Re: Critical and Inflection points of Rational Functions

Thank you Bob. Yes, I mean maxima/ minima. The book I'm using refers to these points as critical points, sometimes extrema as well. I read through the section again and the book does say that there is extrema when f'(c) = 0 or f'(c) = dne. I guess being so tired I wasn't thinking straight. Also, after moving onto the next section it clearly states:

extrema when f'(c) = 0 or f'(c) = dne

and

possible inflection when f''(c) = 0 or f''(c) = dne

Thanks for your time bob.