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Math Help - Continuous function, maximum and minimum

  1. #1
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    Continuous function, maximum and minimum

    I'm exercising for a test I have in two days, and I came across this question from previous exams:

    Let f : (0,1) ---> R be a continuous function. It is given that f does not have maxima nor minima, and that f(0.5) = f(3/4).
    Show that f has a local minimum point in (0,1).

    Any ideas?
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  2. #2
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    Re: Continuous function, maximum and minimum

    Quote Originally Posted by loui1410 View Post
    Let f : (0,1) ---> R be a continuous function. It is given that f does not have maxima nor minima, and that f(0.5) = f(3/4). Show that f has a local minimum point in (0,1).
    Here are some hints. But it still takes some work.
    The high-point/low-point theorem says that any continuous function on a closed interval has a maximum and a minimum.
    [0.5,0,75] is a closed interval. So?
    But you know that f does not have maxima nor minima on (0,1).
    Thus what can you conclude?
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  3. #3
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    Re: Continuous function, maximum and minimum

    That f is a constant function? But anyhow how am I supposed to prove that it has a local minimum point when it says that it has no maxima nor minima? Do they actually mean that it has no global maxima or minima?
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    Re: Continuous function, maximum and minimum

    Quote Originally Posted by loui1410 View Post
    That f is a constant function?
    If the function is constant then it has both a global maxima and minima.

    Quote Originally Posted by loui1410 View Post
    Do they actually mean that it has no global maxima or minima?
    Of course that is exactly what it means. There is a p\in [0.5,0.75] such f(p) is a minimum on that interval.

    If f(p)\ne f(0.5) you are done.

    BUT if f(p)=f(0,5) there is more work to be done.
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