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Upper and Lower Bounds

  1. Jun 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Assume happiness can be measured on a scale from -10 (very unhappy) to 10 (extremely happy). Let D denote the set of possible ways to live one day. For each way of living a day[tex]d∈D[/tex] define a function
    [itex]h_d[/itex][tex]:[0,24]⟶[−10,10][/tex] to be equal to the value of happiness at each point in time. For example, [itex]h_d[/itex][tex](6.25)=−7.3[/tex] means that for this particular way of living a day the level of happiness at 6:15 in the morning is -7.3 (this may happen if one sets alarm clock to 6:00am). Let [itex]H_d[/itex]=glb {[itex]h_d[/itex](t)|{0≤t≤24}. Let H=lub{[itex]H_d[/itex]|d∈D}.

    1) Assuming that happiness changes continuously, prove that for every d∈D there exists t∈ [0,24], such that [itex]h_d[/itex] (t)= [itex]H_d[/itex].

    2) Assume that happiness changes continuously and that H=4. Is it possible to live through a day so that you are always happier than 3.9999999? Is it possible to live through a day so that you are always happier than 4? Justify your answers.

    3) Without assuming that happiness changes continuously, prove that {−10≤H≤10}.


    2. Relevant equations



    3. The attempt at a solution
    Can you walk me through what the question is asking? I've been having difficulty with understanding what exactly is the glb and lub.
     
  2. jcsd
  3. Jun 3, 2013 #2

    tiny-tim

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    Hi Numnum! :smile:

    Each hd is a function, and you can draw it on a 24-hour graph.

    The minimum on that particular graph is Hd.

    The greatest Hd (for all d in D) is H. :wink:
     
  4. Jun 3, 2013 #3
    So for part a) I have to prove that for every [itex]h_d[/itex] function, there exists a minimum? I'm not very good at proofs, so... What would I choose as the function?
     
  5. Jun 4, 2013 #4

    tiny-tim

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    Hi Numnum! :smile:

    (just got up :zzz:)
    I don't understand. :confused:
     
  6. Jun 4, 2013 #5

    Ray Vickson

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    1) is asking you to prove a very standard result, viz., that a continuous function f on a finite closed interval [a,b] attains a maximum and a minimum.
     
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