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Given a function $f(x)$ I can determine whether its concave up or concave down by using the second derivative as it says e. g. here. $$f''(x) > 0 \qquad \text{concave up}$$ $$f''(x) < 0 \qquad \text{concave down}$$

For instance $$f(x) = x^2 \qquad f'(x) = 2x \qquad f''(x) = 2$$

  1. Does that mean that since $f''> 0$ is true for all x, the function $f(x)$ is always concave up?
  2. And how can I determine the quantity of concavity?

I thought somehow that $f''$ gives me the quantity but $f''$ is constant and to me it seems that the concavity of $f$ changes, so I guess this is wrong. I mean $f(x)$ has a stronger concavity around $x = 0$, hasn't it?

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  • $f(x)$ is concave up in its domain. – Mary Star Feb 03 '14 at 07:35
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    For the first question, you are right. But, if the second derivative is still a function of $x$, then it could change sign (this is how arrive inflection points). Plot $y=x^3$ for $-1<x<1$ – Claude Leibovici Feb 03 '14 at 07:36
  • @Marystar What does that mean "in its domain"? –  Feb 03 '14 at 14:18
  • @ClaudeLeib Okay. However, is it only the sign to be considered or does the value ($f''=2$ in case) tell us something, too? –  Feb 03 '14 at 14:20
  • @embert Since the domain of $f$ is the whole $\mathbb{R}$ and $f''=2$ is a positive constant, the function $f(x)$ is concave up in $\mathbb{R}$. – Mary Star Feb 03 '14 at 14:29
  • @marystar so domain is smth like "valid x". And how about the value $2$. Has it any meaning (see my comment above)? –  Feb 03 '14 at 14:32
  • @embert Yes,it's something like that. You have to consider only the sign of $f''$. – Mary Star Feb 03 '14 at 14:34
  • @marystar "You have to consider only the sign of $f''$" for checking quality (up or down) of concavity. Still I'd like to have means to get the quantity.. –  Feb 03 '14 at 14:39
  • @embert In this case where $f''$ is a constant the concavity of $f$ doesn't change. When $f''$ were not contant and you could find some roots of $f''(x)=0$ then you have check the sign 0f $f''$ at each interval between the roots. – Mary Star Feb 03 '14 at 15:25
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    @maryst I had some difficulties with expressing myself. What I was looking for is actually the "curvature".. In this special case $f''$ seems to be the maximum value of the curvature which is in $x=0$ –  Feb 03 '14 at 16:22

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I assume, what I meant with quantity of concavity is actually called Curvature. At least that's the keyword, where I found my "quantity" on wikipedia.

$$cur = \frac{1}{r} = \frac{f''(x)}{(1 + f'(x)^2)^{3/2}}$$

x2 x3

  • It is good you found by yourself what you were looking for. It is much clearer now. Do the same with more complex functions for which the second derivative is not a constant ($y=x^3$ for example). Cheers. – Claude Leibovici Feb 04 '14 at 09:49
  • @claudeleibovici yea, things are a bit clearer now. Is it coincidence, that for $f(x)= x^2$ the constant $f''$ corresponds to the maximum curvature or is there any other illustrative meaning of the values of $f'' (x)$ in general (besides its sign indicating up/down concavity)? –  Feb 04 '14 at 10:15