The Concavity: The large chunk of New England (and parts of southeastern Québec) made uninhabitable by United States waste disposal and the annularized fusion developed to deal with said waste. This process resulted in uncontrollable rapid cycles of growth and decay, rampaging packs of feral hamsters, and giant (also feral) skull-less babies. This land was "gifted" to Canada as part of the O.N.A.N. deal. Known to Quebequois (especially séparatistes) as The Convexity.(IJ)

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.

Con*cav"i*ty (?), n.; pl. Concavities (#). [L. concavitas: cf. F. concavit'e. See Concave.]

A concave surface, or the space bounded by it; the state of being concave.

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