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Math Help - Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

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    Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

    Assume that f is a differentiable function such that f(0)=f'(0)=0 and f''(0)>0. Argue that there exists a positive constant a>0 such that f(x)>0 for all x in the interval (0,a). Can anything be concluded about f(x) for negative x's? My teacher hinted at using the Mean value theorem but I can never seem to get MVT problems right.
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    Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

    Quote Originally Posted by NWeid1 View Post
    Assume that f is a differentiable function such that f(0)=f'(0)=0 and f''(0)>0. Argue that there exists a positive constant a>0 such that f(x)>0 for all x in the interval (0,a). Can anything be concluded about f(x) for negative x's? My teacher hinted at using the Mean value theorem but I can never seem to get MVT problems right.
    The conditions given imply that there is a minimum at the origin. So that means that to the right of the origin, the function must increase, even if just for a little while.
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    Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

    Quote Originally Posted by NWeid1 View Post
    Assume that f is a differentiable function such that f(0)=f'(0)=0 and f''(0)>0. Argue that there exists a positive constant a>0 such that f(x)>0 for all x in the interval (0,a).
    Use the limit definition of derivative.
    We know that f"(0) exists and is positive.
    Therefore, \left( {\exists \delta  > 0} \right)\left( {\forall x \in (0,\delta)}\right)\left[ {\left| {\frac{{f'(x) - f'(0)}}{{x - 0}} - f''(0)} \right| < \frac{{f''(0)}}{2}} \right].

    From which we get \left( {\forall x \in (0,\delta )} \right)\left[ {f'(x) > x\frac{{f''(0)}}{2}} \right].
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    Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

    Plato: I haven't learned this method :\
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    Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

    Quote Originally Posted by NWeid1 View Post
    Plato: I haven't learned this method :\
    Are you saying that you never learned the definition of a derivative?
    How can you be asked to do such a question then?
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    Re: Assume f(0)=f'(0)=0, prove there exists a positive constant such that f(x)>o...

    Not the formal definition. My teacher said to use mean value theorem
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