1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Minimum number of real roots

  1. Oct 16, 2013 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Let f(x) be a non-constant twice differentiable function defined on R such that f(x)=f(1-x) and f'(1/4) =0 then what is the minimum number of real roots of the equation (f"(x))^2+f'(x)f'''(x)=0.

    3. The attempt at a solution


    putting x = 1/4 in the first eqn gives f'(3/4)=0
    putting x=1/4 in the given equation (f"(x))^2+f'(x)f'''(x)=0 gives f"(1/4)=0. ∴f"(3/4)=0.
    So I have got a total of 2 roots. But there are some more which I can't find.
  2. jcsd
  3. Oct 17, 2013 #2
    Observe that the given equation is
    $$\frac{d}{dx}(f'(x)\cdot f''(x))=0$$
    See if that helps.
  4. Oct 17, 2013 #3


    User Avatar
    Gold Member

    Let y= f'(x)f"(x)

    According to Rolle's Theorem there must lie a root of dy/dx between 1/4 and 3/4. But what about other roots?
  5. Oct 17, 2013 #4
    I am unsure how to proceed but how about x=1/2?

    At how many points is f'(x) zero? At how many points is f''(x) zero?
    Last edited: Oct 17, 2013
  6. Oct 17, 2013 #5


    User Avatar
    Homework Helper

    You need to find the known zeroes of [itex]y = f'f''[/itex], at which point you can apply Rolle's theorem to get the known zeroes of [itex]y'[/itex], which is what you're interested in.

    So far you know that [itex]f'(\frac14) = f'(\frac34) = 0[/itex].

    What, in view of the condition [itex]f(x) = f(1-x)[/itex], can you say about [itex]f'(\frac12)[/itex]?

    Now apply Rolle's theorem to [itex]f'[/itex].
  7. Oct 18, 2013 #6


    User Avatar
    Gold Member

    I can say that there must lie a root of y' between 1/4 and 1/2 as well as 1/2 and 3/4.
  8. Oct 18, 2013 #7
    y is also zero at some more points.

    f'(x) is zero at 1/2 and 1/4 so there must be at least one point in between them where f''(x) is zero. Can you proceed now?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted