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Math Help - absolute maximum in [a,b)

  1. #1
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    absolute maximum in [a,b)

    Suppose that the function f is continuous on [a,b) and the limit L=\lim_{x\rightarrow{b^-}}{f(x)} exists.
    (i) Prove that if there is an x_0\in{[a,b)} such that f(x_0)>L, then f has an absolute maximum in [a,b).
    How to approach this question?

    (ii) If there is an x_1\in{[a,b)} such that f(x_1)=L, does f necessarily have an absolute maximum in [a,b)?
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  2. #2
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    Re: absolute maximum in [a,b)

    I'm assuming L is a finite real number, if that's the case then

    (i) By the definition of limit there is an s>0 such that if 0<|x-b|<s then f(x)-L\leq |f(x)-L|< f(x_0)-L so that f(x)<f(x_0) on (b-s,b). From here it should be obvious.

    (ii) If we define f(b)=L then f is continous on [a,b], by the extreme value theorem we have an absolute maximum M, then M\geq L. Consider the cases M=L, M>L.
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  3. #3
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    Re: absolute maximum in [a,b)

    Quote Originally Posted by Jose27 View Post
    I'm assuming L is a finite real number, if that's the case then

    (ii) If we define f(b)=L then f is continous on [a,b], by the extreme value theorem we have an absolute maximum M, then M\geq L. Consider the cases M=L, M>L.


    This method can also be adapted to do part (i) thus saving work by using the same idea for both.

    CB
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  4. #4
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    Re: absolute maximum in [a,b)

    Last edited by alphabeta89; December 14th 2011 at 01:08 AM.
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