# Concave/convex -- second derivative

1. Dec 4, 2015

### charlies1902

Hello. I have a question regarding curvature and second derivatives. I have always been confused regarding what is concave/convex and what corresponds to negative/positive curvature, negative/positive second derivative.
If we consider the profile shown in the following picture:
https://www.princeton.edu/~asmits/Bicycle_web/pictures/vel_profile.GIF

U(y) is the velocity plotted as a function of (y).
We can see that the second derivative here is <0. DOes this correspond to negative curvature or positive and concave/convex?

2. Dec 4, 2015

### Samy_A

Positive second derivative corresponds to convex (you can figure this as the tangent being below the graph of the function near the point).
Negative second derivative corresponds to concave (you can figure this as the tangent being abovethe graph of the function near the point).

The function in your picture has a positive second derivate.

3. Dec 5, 2015

### Staff: Mentor

The terminology used to be "concave up" (like the graph in the image) versus "concave down." A graph that was concave up over an interval has a positive second derivative on that interval, while one that was concave down over an interval had a second derivative that was negative over that interval.

Derivate is a word, @Samy_A, but I'm sure the one you meant is derivative.

4. Dec 5, 2015

### Samy_A

Of course, sorry for the mistake.

5. Dec 5, 2015

### Staff: Mentor

As a remark from someone who tends to mix left and right, concave up and down and so on, all direction notations that depend on the point of view, it often helps me out to consider the standard parabolas $± x^2$.

6. Dec 5, 2015

### Staff: Mentor

Concave up -- "holds water"
Concave down -- "water runs out"

With regard to left/right, after US Army Basic Training, I was very clear on left and right...

7. Dec 5, 2015

### Staff: Mentor

There's an interesting question my algebra professor asked in his very first lecture which might be funny to post it here somewhere, too.
"Why does a mirror change left and right but not up and down?" With respect to, e.g. particle physics, this could be a good exercise when it comes to broken symmetries. (Just a thought.)

8. Dec 9, 2015

### mathexam

Yeah, concavity is the second derivative. If f''(x) > 0, the function is concave up. If f"(x) < 0 the function is concave down.

9. Dec 10, 2015

### Staff: Mentor

That's what I said in post #3.