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Intuition for positive third derivatives

  1. Jun 16, 2014 #1
    We can get a lot of good intuition for how first and second derivatives work by interpreting a sign restriction.

    Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R##.

    1) If ##f## is differentiable, then ##f## is monotone iff ##f'\geq 0## everywhere.
    2) If ##f## is twice differentiable, then ##f## is convex iff ##f''\geq 0## everywhere.

    Monotonicity and convexity are very nice for a couple reasons:
    - They're easy to interpret/visualize. e.g. Convexity means all secants lie above the graph.
    - They're both global properties.
    - They're both easy to state without having defined a derivative.

    Is there a similarly interpretable condition which is equivalent (when ##f## happens to be thrice differentiable) to ##f'''\geq0##? Ideally, I'd be interested in a condition which (like monotonicity and convexity) is global and makes no reference to differentiation.
     
    Last edited: Jun 16, 2014
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  3. Jun 16, 2014 #2

    Matterwave

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    If f'>0 everywhere, then f is increasing everywhere (can start negative and go positive). If f''>0 everywhere then f' is increasing everywhere (can be negative and go positive). If f'''>0 everywhere, then f'' is increasing everywhere (can be negative and go positive), and so on.

    As you get higher and higher up, you get conditions on the f itself that is further removed from f.

    So, if f'''>0 and this is the only restriction, then the function can start out concave and turn convex, once convex, it can only remain convex, but it cannot start out convex and turn concave. I don't know if this is intuitive to you because really the restriction is harder to even see on a graph.

    Maybe if you think of a polynomial of degree 3: ##y=ax^3+bx^2+cx+d##. In this case f'''>0 specifies that a>0 (and only that a>0, it puts no restrictions on any other terms). In this case, there can only ever first be a hump and then a valley, it can't be valley first then hump.

    On a 4th degree polynomial, for example, you can't have the characteristic 2 bumps with a valley separating them that one is used to seeing on a 4th degree polynomial, because that function would go convex to concave to convex. Etc.
     
  4. Jun 16, 2014 #3
    But a person doesn't need to know/care about how ##f'## is defined for me to explain the condition that "##f''\geq 0##". Indeed, I can describe it as follows: if any line segment has both endpoints above the graph of ##f##, then the whole line segment lies above the graph of ##f##. This is, aside from twice differentiability, the full content of ##f''\geq 0##.

    I'm wondering whether there's a similarly geometric way of describing the condition ##f'''\geq 0##.
     
  5. Jun 16, 2014 #4

    Matterwave

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    "The function can only turn from concave to convex, never in reverse." This is not a geometric way of describing f'''>0?
     
  6. Jun 17, 2014 #5
    That's geometric, but it doesn't characterize ##f'''\geq 0##. For instance, the smooth function ##f:\mathbb R\to\mathbb R## given by ##f(x)=x^4## never turns from convex to concave (as it's globally convex), but it doesn't satisfy ##f'''\geq 0##.
     
  7. Jun 17, 2014 #6

    Matterwave

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    So you want a if and only if statement for f'''>0 that is geometric and intuitive? Sorry, but unless you can intuitively understand "the convexity of the graph must always be increasing"...I don't know of any other such statements.
     
  8. Jun 17, 2014 #7

    AlephZero

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    You could look at this question from a different angle. The "simplest" visual ways to describe a curve are probably slope and curvature. Those idea map onto the notions of first and second derivatives in a fairly simple qualitative way. (Qualitative but not quantitative - for example the radius of curvature is a messy formula involving both first and second derivatives).

    So, find the next simplest way to visualize a curve, and see it you can map it onto the third derivative. A good choice might be "spirality", i.e. how the curvature changes as you move along the curve, and whether it is "coiled up" more at one end than the other....
     
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