1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Applications of Derivative - Find no. of roots of

  1. Aug 6, 2012 #1

    AGNuke

    User Avatar
    Gold Member

    if f(x) is twice differentiable function such that f(a)=0; f(b)=2; f(c)=-1; f(d)=2; f(e)=0, where a<b<c<d<e; then minimum number of zeroes of g(x) = (f'(x))2+f''(x)f(x) in the interval [a,e] is ......


    All I can figure out is that at the least, it is a 4-degree polynomial with roots a, (b,c) (a root in between b and c), (c,d), e.
     
  2. jcsd
  3. Aug 6, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Here's a hint. What's the second derivative of f(x)^2?
     
  4. Aug 6, 2012 #3

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    f need not be a polynomial
    f must have at least four roots. for eg. in the interval [b,c], f must cross the x-axis any odd number of times ;)

    So sketch the least-zeros case - just guess.
    For the same guess, sketch the first and second derivative. You'll see that f' is zero at turning points (and points of inflection - but you want least zeros, so your guess for f should probably avoid inflections) and f'' is zero when there are turning points in f'.

    Notice that any number multiplied by zero is zero.
    Compare your values with g(x).
     
  5. Aug 7, 2012 #4

    AGNuke

    User Avatar
    Gold Member

    This is my attempt. My mind was too fogged due to illness that I can't see something.

    [tex]g(x)=\frac{\mathrm{d} }{\mathrm{d} x}(f(x).f'(x))[/tex]

    Integrating g(x) with respect to x.

    [tex]G(x)=f(x).f'(x)[/tex]

    Now G(x) has minimum 7 zeroes (4 for f(x); 3 for f'(x)); therefore, g(x) must have minimum 6 zeroes.
     
  6. Aug 7, 2012 #5

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    That may not be the minimum for G(x) if f has a turning point at a zero.
    This is part of why I suggested sketching f(x).

    If you integrate an order 4 polynomial with 4 distinct zeros, do you usually get fewer zeros? Don't you normally reduce the order by differentiating?

    In fact - looking at g(x) - it seems it may have trivial zeros where f'(x) and either f(x) or f''(x) are zero. ie either f(x) has a turning point where f(x)=0 or f'(x) has a turning point where f'(x)=0. Is this the case?

    Non-trivially, (f'(x))^2 is always positive, while f''(x).f(x) may be positive or negative ... so there are likely to be some places where f''(x).f(x)=-(f'(x))^2
     
  7. Aug 7, 2012 #6

    AGNuke

    User Avatar
    Gold Member

    I sketched f(x). Roots of f(x) are a, (b,c), (c,d), e. I can think of a graph similar to a 4 degree polynomial.

    I simply deduced G(x) which is ∫g(x).dx Looking upon the question, Since it is asking for minimum roots, f(x) is set to 4 degree polynomial, at least can be assumed for the given interval. We only have to find the roots of g(x) in [a,e], so 4 roots (minimum) of f(x) in the interval. So I think there should be no inflection points there.

    As of f'(x), Since there are no inflection points, there must be (4-1)=3 zeroes, each between successive zeroes of f(x).

    G(x) = f(x).f'(x) have 7 zeroes at minimum. Therefore, differential of G(x), which is g(x) have 6 zeroes.

    I see nothing wrong in it... :frown:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Applications of Derivative - Find no. of roots of
Loading...