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Monotonically decreasing/increasing

  1. Oct 11, 2014 #1
    1. The problem statement, all variables and given/known data

    Plot each function and decide, based on the plot whether or not it is monotonically increasing/decreasing or strictly monotonically increasing/decreasing.

    f(x) = -5

    2. Relevant equations


    3. The attempt at a solution

    I can plot this function no problem, and show algebraically why it fits the definition of monotonically increasing and monotonically decreasing.

    My question is, how do I justify this 'based on the plot'?

    The next part of the question is to show it algebraically, but graphically i'm not really sure why my answer is correct...

    Thanks for any help you can give.
     
  2. jcsd
  3. Oct 11, 2014 #2

    Ray Vickson

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    You really need to show more: what exact definitions are you using for "monotonically increasing/decreasing"? You seem to be making a distinction between monotonically increasing and strictly monotonically increasing, etc. Your answer could be right or wrong, depending on these details.
     
  4. Oct 11, 2014 #3

    Mark44

    Staff: Mentor

    The question distinguishes between monotonically increasing/decreasing and strictly monotonically increasing/decreasing. In the latter, the graph of the function would have to be trending up (mon. increasing) or down (mon. decreasing). From your plot, which should show a horizontal line, which you could characterize as both monotically decreasing and monotonically increasing.
     
  5. Oct 11, 2014 #4
    I have attached the definitions I am using as a picture. I will explain how i'm using them to come to my result.

    flR77zx.png

    From this, I say that the function [itex]f(x) = -5[/itex] is monotonically increasing and monotonically decreasing because for all [itex]x_{1}, x_{2} \in \mathbb{R}[/itex] where [itex]x_{1} < x_{2}, f(x_{1}) \leq f(x_{2})[/itex].

    Likewise, for all [itex]x_{1}, x_{2} \in \mathbb{R}[/itex] where [itex]x_{1} < x_{2}, f(x_{1}) \geq f(x_{2})[/itex].
     
  6. Oct 11, 2014 #5

    Mark44

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    I figured that these were the definitions you were using. I agree with your answer.
     
  7. Oct 11, 2014 #6

    vela

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    Is your question essentially "How can a flat graph be considered increasing or decreasing?"
     
  8. Oct 11, 2014 #7
    that is at the heart of it.
     
  9. Oct 11, 2014 #8

    vela

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    It's just how it is. As long as the function satisfies the definition, we say it's monotonically increasing or decreasing, even if it may run counter to our everyday sense of the words. Your intuition aligns with strictly monotonically increasing or decreasing.
     
  10. Oct 11, 2014 #9

    Mark44

    Staff: Mentor

    Note that the definitions for monotonically increasing and decreasing include the possibility that f(x1) = f(x2). IOW, they include the possiblity of graphs that are flat. The "strictly" definitions don't include this.
     
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