Concavity is a
wacky and fun
calculus concept.
Actually, it's just a calculus concept on which I have a test tomorrow.
A function can be said to be either concave up or concave down. As Mr. Bergen explains it, a curve that is concave up can hold water. Example: y=x² . One that is concave down, obvioiusly, can't. Example: y=-x² . (That's -(x²), not (-x)², so don't go complaining that I'm wrong.)
The easiest way to determine the concavity of a function f(x) is to use the second derivative test. If you don't know what a second derivative is, you probably have no real need to mathematically determine concavity.
If the second derivative at a constant is positive, the function is concave up. If it's negative, the function is concave down. If it's zero, it's a point of inflection, a point where the curve goes from concave up to concave down or vice versa.*
Here's an example:
f(x)=4x³+3x²+6x+4
f'(x)=12x²+6x+6
f''(x)=24x+6
f''(x)=0 when x=-1/4
Therefore, x=-1/4 is a point of inflection. When x<-1/4, f''(x)<0, so f(x) is concave down there. When x>-1/4, f''(x)>0, so f(x) is concave up.
*If f''(x)=0, there's still a chance that it won't be a change in concavity, that it will just be a screwy bit in the function. So be sure to test to the left and right to make sure. I can't think of an example of this, so if you can provide one or prove me wrong, /msg me.
Also, if you have experience in high-level math, don't suggest that I mention such-and-such type of an exception. It's October. I'm in BC Calculus. I don't know that yet.