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Just wondering why knowing the concavity of a graph matters. I see just how, and not why. The how is "mathematical" and academic. Practically speaking, if I had a graph of my height over time, what do I care about the concavity? What does it tell me about my height?

The 1st derivative tells us where the graph is increasing/decreasing. The 2nd derivative tells us where the concavity is up/down.

When f'' is (+), it means f' is increasing. That means the rate of change of f is increasing. So, the change is accelerating, so to speak?

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    $\begingroup$ Concave/convex functions are hugely useful when doing estimates. An important tool is Jensen's inequality. $\endgroup$
    – J.R.
    Commented Mar 12, 2015 at 16:40
  • $\begingroup$ This all seems too "mathematical" and academic. If I had a graph of my height over time, what do I care about the concavity? What does it tell me about my height? $\endgroup$
    – JackOfAll
    Commented Mar 12, 2015 at 18:13

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As you mentioned in the question body, concavity indicates acceleration/deceleration.

Imagine a distance-time graph of a car. If the graph is a straight diagonal line, that indicates the car is traveling at a constant speed; the distance traveled increases at a constant rate.

If the graph is a curved line which becomes more steep as time progresses (concave up), that indicates the car is speeding up, allowing the car to cover more distance in each successive equal interval of time. Taking it a step further, we can infer that there must be a force that causes the car to speed up, and the force must be acting in the same direction that the car is moving (as stated in Jack D'Aurizio's comment).

Similarly, if the distance-time graph is a curved line which becomes less steep as time progresses (concave down), that indicates the car is slowing down, which is why it covers less and less distance for each successive equal time interval. Again, this implies a force acting on the car, but this time in the opposite/negative direction of the car's motion.

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  • $\begingroup$ Yes, this is what I needed back then! $\endgroup$
    – JackOfAll
    Commented Jan 25 at 14:14
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Concavity/convexity is a great source of useful inequalities.

For instance, any convex function $f$ on $[a,b]$ satisfies: $$ f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_{a}^{b}f(y)\,dy \leq\frac{f(a)+f(b)}{2}$$ also known as Hermite-Hadamard's inequality.

A straightforward consequence is: $$ \frac{1}{x+\frac{1}{2}}\leq \log\left(1+\frac{1}{x}\right)\leq\frac{1}{2}\left(\frac{1}{x}+\frac{1}{x+1}\right)$$ that is really useful when providing bounds for $\frac{n!e^n}{n^n}$.

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  • $\begingroup$ This all seems too "mathematical" and academic. If I had a graph of my height over time, what do I care about the concavity? What does it tell me about my height? $\endgroup$
    – JackOfAll
    Commented Mar 12, 2015 at 18:13
  • $\begingroup$ @JackOfAll: in a more concrete way, if $y(t)$ is the position of a particle at time $t$, then the concavity of the graphics gives you the sign of the force that leads the motion, since $F=m\cdot y''(t)$. $\endgroup$
    – Jack D'Aurizio
    Commented Mar 12, 2015 at 18:42
  • $\begingroup$ LOL, you lost me. $\endgroup$
    – JackOfAll
    Commented Mar 14, 2015 at 13:49
  • $\begingroup$ Was hoping for one simple example where concavity means something obvious. $\endgroup$
    – JackOfAll
    Commented Mar 14, 2015 at 13:50

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