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A question about the minimum/maximum of a convex function

  1. Nov 8, 2005 #1
    I would like to be sure in the following, not prove it, just have it confirmed...
    If a function f is convex, then it has

    1.) only one maximum and no minimum
    2.) only one minimum and no maximum

    infinity and -infinity are not included.
     
  2. jcsd
  3. Nov 8, 2005 #2

    HallsofIvy

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    It's not clear what you are asking. Since you say you want something "confirmed" it would appear you are making a statement: that either one of those two statements can be true for a convex function. That's not correct.

    A convex function is a function f, such that the set {(x,y)| y> f(x) } is convex. Given that, on (2) is true.
    (1) is true for a concave function.
     
  4. Nov 8, 2005 #3
    Oh, ok, I didnt make a clear distinction between a convex and a concave function.

    I didnt have a clear definition of convex and concave, so it makes more sense now, given your definition, and concave is then the opposite, and so makes (1) true.

    thanks!
     
  5. Nov 8, 2005 #4

    HallsofIvy

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    In particular, y= x2 is a convex function (the set of point above its graph is convex) and y= -x2 is a concave function (the set of points above its graph is concave).
     
  6. Nov 13, 2005 #5
    Convex...I knew it's concave up or concave down. Well, imagine the graph of y=x^2 or y=-x^2 depending on what you're talking about. It's a simple visualisation any Algebra 2 student can do...

    bah...the guy above me wrote this exact thing.

    However, I wonder, is there such a thing as Convex?
     
  7. Nov 13, 2005 #6

    benorin

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    yes. In terms of functions: convex = concave up & concave = concave down.

    I'm guessing that the whole up/down stuff is from a Calculus text by Stewart.
     
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