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Math Help - Prove

  1. #1
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    Prove

    Suppose f,g, & h are defined on (a,b) & a<x0<b. Assume f and h are differentiable at x0, f(x0)=h(x0), & f(x)<g(x)<h(x) for all x in (a,b). Prove that g is differentiable at x0 and f'(x0)=g'(x0)=g'(x0).
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  2. #2
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    Re: Prove

    Quote Originally Posted by franios View Post
    Suppose f,g, & h are defined on (a,b) & a<x0<b. Assume f and h are differentiable at x0, f(x0)=h(x0), & f(x)<g(x)<h(x) for all x in (a,b). Prove that g is differentiable at x0 and f'(x0)=g'(x0)=g'(x0).
    It is important that f(x_0)=h(x_0) so -f(x_0)=-h(x_0).

    If \delta>0 can you show that
    \frac{f(x_0+\delta)-f(x_0)}{\delta}<\frac{g(x_0+\delta)-g(x_0)}{\delta}<\frac{h(x_0+\delta)-h(x_0)}{\delta}~?

    There is a similar statement if \delta<0.

    Those are used on the limit definition of the derivatives and f'(x_0)=h'(x_0).
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