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Real Analysis : Points on a Differentiable Function

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Let h be a differentiable function defined on the interval [0,3], and assume that h(0)=1 h(1)=2 and h(3)=2.
a- argue that there exists a point d belong to [0,3] where h(d)=d.
b-argue that at some point c we have h'(c)=1/3.
c-argue that h'(x)=1/4 at some point in the domain.

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Points on a differentiable function are investigated in the solution.

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a. Let f(x)=h(x)-x. Then we have f(0)=h(0)-0=1-0=1>0, f(3)=h(3)-3=2-3=-1<0. According to the Intermeidate Value Theorem, we can find some d in [0,3], such that f(d)=0. This implies h(d)-d=0, so ...

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