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Math Help - prove there is a c_1

  1. #1
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    prove there is a c_1

    Suppose f: [0,2]-> R is differentiable. f(0)=0, f(1)=2 and f(2)=2. Prove that

    1. There is c_1 such that f'( c_1) =0
    2. There is c_2 such that f'(c_2)=2 and
    3. There is c_3 such that f'( c_3=1/2
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  2. #2
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    For (1) and (2), use the mean value theorem on the intervals [1,2] and [0,1] respectively. Since a derivative always has the intermediate value property (Darboux's theorem), it follows that there is a c_3 between c_1 and c_2 where the derivative is 1/2.
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  3. #3
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    Quote Originally Posted by maddas View Post
    For (1) and (2), use the mean value theorem on the intervals [1,2] and [0,1] respectively. Since a derivative always has the intermediate value property (Darboux's theorem), it follows that there is a c_3 between c_1 and c_2 where the derivative is 1/2.
    Hmm..that makes sense lol thanks !
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