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Basic question: First derivative test to detect whether a function is decreasing

  1. Jun 2, 2009 #1
    If the first derivative of a function f from R to R is negative on [a,b], it IS right to say that the function is decreasing on [a,b] right?

    Are there any other ways of showing that the function is decreasing on [a,b]?
     
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  3. Jun 2, 2009 #2

    tiny-tim

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    Hi seeker101! :smile:
    Yes.
    No … for a differentiable R->R function, negative derivative at a point is the same as decreasing. :wink:
     
  4. Jun 2, 2009 #3

    EnumaElish

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    That's correct as a definition. But, what the OP is asking may be whether there are other ways of arguing that a function is decreasing over an interval.
     
  5. Jun 2, 2009 #4

    nicksauce

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    Of course there are other ways to show a function is increasing (the same argument can be applied to decreasing) on an interval. For example, using the definition of increasing: if x,y are in [a,b] then show y > x implies f(y) >= f(x).

    Example: Show that f(x) = x^2 is increasing on [0,2].
    Let x and y belong to [0,2] with y > x. Then we can write y = x + e, for some e>0. Then f(y) = (x+e)^2 = x^2 + 2xe + e^2 > f(x) = x^2, since e^2 is >0 and 2xe is >0.
     
  6. Jun 2, 2009 #5
    how about a function f is decreasing on [a,b] if [tex](\forall x_1,x_2\in [a,b])[/tex] [tex]x_1<x_2 \implies f(x_1) \ge f(x_2)[/tex]
     
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