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Okay, a stupid question about curve sketching

  1. Jun 27, 2006 #1
    Is it possible to have 2 global minimums? I'm just having trouble determining whether this quartic has minimums or not =/
     
    Last edited: Jun 27, 2006
  2. jcsd
  3. Jun 27, 2006 #2

    NateTG

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    No. There is only one global minimum, however, a function can be minimal in more than one place.

    For example, the function:
    f(x)=0
    is minimal everywhere.
     
  4. Jun 27, 2006 #3
  5. Jun 27, 2006 #4

    Hurkyl

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    (0, -2) isn't a local minimum.
     
  6. Jun 27, 2006 #5
    err...it should be maximum, right?
     
  7. Jun 27, 2006 #6

    Hurkyl

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    Right! (I wasn't sure if you were marking it as a minimum or not, but I wanted to be sure you noticed)
     
  8. Jun 27, 2006 #7
    That was a typo on my part (thank you for noticing it!)
    So there are no minimums in this case?

    Becaue when I try to calculate it, the 3 critical numbers I get are 2,-2, and 0. But if I sub in 2 or -2, I get 6, which doesn't seem right...
     
  9. Jun 27, 2006 #8
    No, there are minimums, just no absolute minimums. There are actually 2 local minimums, and one local maximum between them.
     
    Last edited: Jun 27, 2006
  10. Jun 27, 2006 #9
    But isn't the definition of the minimum (not at a domain endpoint) that:

    [tex]f(x \pm \epsilon) > f(x)[/tex] for sufficiently small [tex]\epsilon[/tex]

    But [tex]f(x \pm \epsilon) = f(x)[/tex] if [tex]f(x)=0[/tex] for all x and so would not have any minimum.
     
    Last edited: Jun 27, 2006
  11. Jun 27, 2006 #10

    shmoe

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    Your plot should make it obvious that there is in fact a global (absolute) minimum so you should either distrust your plot or your work.

    Check your critical points again! (in the plot we trust)

    Nope, it's a less than or equal to, [tex]\leq[/tex], for a minimum. Or [tex]\geq[/tex] if you're looking in a mirror.
     
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