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Math Help - Need help for 2 interesting questions - rolle's and continuity

  1. #1
    owq
    owq is offline
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    Need help for 2 interesting questions - rolle's and continuity

    1. Let f and g be functions such that f'' and g'' exist everywhere on R. For a < b, suppose that f(a)=f(b)=g(a)=g(b)=0, and g''(x) \not= 0 for every x \in (a,b).

    (i) Prove that g(x) \not= 0 for every x \in (a,b). *I know how to do this, if I'm not wrong, use Rolle's Theorem a few times.*

    (ii) Show that there exists a number c \in (a,b) for which \frac{f(c)}{g(c)} = \frac{f''(c)}{g''(c)} *No idea. Does it have something to do with the Mean Value Theorem?*

    2. Let f be a continuous function on R such that \lim_{x\to0}\frac{f(x)}{x} exists. Define the function
    g(x) = \int_0^1 f(xt) dt, x \in R
    Determine whether g' is continuous at x = 0. Justify your answer.
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    Quote Originally Posted by owq View Post
    1. Let f and g be functions such that f'' and g'' exist everywhere on R. For a < b, suppose that f(a)=f(b)=g(a)=g(b)=0, and g''(x) \not= 0 for every x \in (a,b).

    (i) Prove that g(x) \not= 0 for every x \in (a,b). *I know how to do this, if I'm not wrong, use Rolle's Theorem a few times.*

    (ii) Show that there exists a number c \in (a,b) for which \frac{f(c)}{g(c)} = \frac{f''(c)}{g''(c)}
    The function f(x)g'(x) - f'(x)g(x) vanishes at both ends of the interval, so ..... (Rolle).
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