Quadratic equations are termed concave up when U-Shaped and concave down when |¯| shaped. These shapes occur when the x² coefficient is positive and negative, respectively.
(See the 2 graphs on this page: http://www.1728.com/minmax.htm)
Moving on to cubics, (see graph http://www.1728.com/minmax2.htm), the curve takes one more turn and so, when graphed, cubic equations can
Start from negative infinity and go to positive infinity (as shown) OR can
Start from positive infinity and go to negative infinity.
In the case of cubics, what are these shapes called and what determines if they are like case 1 or case 2?
Quartic Equations (not shown) take one more turn and so the curves can go from positive infinity to positive infinity OR negative infinity to negative infinity in which case the curves could be roughly described as
‘w’ shaped or ‘m’ shaped, respectively.
In the case of quartics, what are these shapes called and what determines if they are ‘w’ shaped or ‘m’ shaped?
Let p[sub]n/sub denote an nth degree polynomial, and a[sub]n[/sub] the coefficient of x[sup]n[/sup] in p[sub]n/sub. As |x| [symbol]® ¥[/symbol], p[sub]n/sub/x[sup]n[/sup] [symbol]®[/symbol] a[sub]n[/sub]. So it’s the sign of the leading coefficient that determines the behavior of the polynomial for large |x|. In other words, it’s just like the quadratic case.
Concavity is defined locally – at each point – by taking the second derivative. Consider y = ax[sup]2[/sup] + bx + c. The second derivative is 2a, which shares the sign of a, and is the same everywhere.
Now given a general function y = f(x), the second derivative f’’(x) is either positive, negative, or zero at a given value of x. If positive (like the second derivative of a concave-up quadratic) then the curve is concave-up at that value of x. Similarly if it’s negative for some x-value, the curve is concave-down there.
Rereading, I think you were asking partly about global shapes, not just concavity. There in general are no standard terms for such global shapes, and the formulas in terms of coefficients for when a quartic curve is “w-shaped” or “m-shaped” (once you’ve determined whether it goes up or down at the ends as ultrafilter indicates) get a little hairy. What I can tell you is that odd-degree polynomials will go different directions at either end (one up, one down), while even-degree ones will go the same direction. In between will be some number (call it k) of “humps”, the most being n-1 (since a polynomial has at most n distinct real roots) and k will differ from n-1 by an even number. These will alternate up and down, giving k/2 humps in each direction for an odd-degree and one less in the direction both ends go for even-degree.
Given a general function y = f(x), the second derivative f"(x) may not be defined.
Given a polynomial function y = f(x), the second derivative does indeed exist at all points in the domain of f. More generally, all derivatives fSUP/SUP are defined at all points in the domain of f.
Is concavity defined where f’’ vanishes? I’m thinking, for instance, of the function f(x) = x[sup]4[/sup] at the origin. Can one define the concavity in terms of nearby points?
And how about those points where f’’ is not defined? It seems to me that there ought to be some definition of concavity which could accommodate the absolute value function, for instance. It certainly seems like abs(x) is concave up at the origin; is there any way to rigorously say this?
Concavity can only be reasonably defined at all in terms of the tangent line. No tangent line, no concavity. The standard definition is in terms of curvature, as I laid out.
For what you’re getting at… I don’t think there’s a sensible way of doing it. Let y be the square root of the inverse function of x[sup]3[/sup]+x. The second derivative test says that this function is concave-down for every value of x other than zero, but at zero it looks exactly (to first order) like the absolute value function, which you’d say should be concave up. Strange indeed.
The problem is that concavity is really a property of the germ of the function. On this view, yes the notion can be sensibly extended to functions like y = x[sup]4[/sup] by taking a suitable étale topology on the sheaf of germs of C[sup]1[/sup] R-valued functions on R. Then the second-derivative test gives an open set of germs with a well-defined concavity for each value of concavity, and we can extend that notion of concavity to the boundary of each region since no germ can be in the boundary of both regions.
I kinda remember one of my professors talking about convex function theory as being able to deal with non-differentiable functions, but I never got any details.
Saddle points only occur in multivariable functions. A point of inflection is where the second derivative changes sign, and thus the function changes concavity.
Actually, I just thought of another definition that’s simpler to explain, but doesn’t seem as general as the other one.
A function is concave up at x if there are left and right neighborhoods of x such that for any point x[sub]1[/sub] in the left neighborhood and x[sub]2[/sub] in the right neighborhood, the function lies below the secant line between x[sub]1[/sub] and x[sub]2[/sub]. Concave down replaces “below” by “above”.
If “left neighborhood” and “right neighborhood” mean what I think they do, then that definition would, in fact, be able to accomodate discontinuities in derivative. For the absolute value function, for instance, one could choose the left and right neighborhoods as (-1,0] and [0,1) (or possibly (-1,0) and (0,1), depending on the exact definition of left and right neighborhoods), and thereby conclude that the absolute value function is concave upwards at zero. Under this definition, though, the absolute value function would still have no concavity at other points, and your example function would still be concave downward at all other points, since one could choose points in each neighborhood which are on the near side of zero.
This is true, and gives the “expected” answers for all points. Really it’s inspired by the notion of a subharmonic function in complex analysis. I haven’t verified that it matches the sheaf of germs definition, though.
Mathochist Posting #4
Given that all odd-degree polynomials will have values that go from negative infinity to positive infinity (or vice versa), couldn’t such polynomials be classified as ‘/’ or ‘’ ? For example, the graph at this site: http://www.1728.com/minmax2.htm
could be classified as a ‘/’ type of curve.
Yes that is a vague symbol to use but is there a specific term for it?
Moving to even-degree polynomials, those also have two cases
NEVER goes to negative infinity (roughly concave UP) and
NEVER goes to positive infinity (roughly concave DOWN).
Any specific mathematical terms for these 2 cases?
(Just trying to get this topic back down to a level where I can understand it.)
I think all the OP’s other questions have been addressed (to the OP’s satisfaction?—let us know if you’re still confused), so I’ll just chime in and say that, to the best of my knowledge, there is no special name for the shape of the graph of a third or higher degree polynomial the way the graph of a quadratic is called a parabola.
Not really. The even cases are summed up by saying that the function is bounded below or bounded above. In general, though, you just say what direction the function goes at high positive or negative values of x. It’s really a very small amount of information, but that’s really all the information there is in parabolæ anyhow.
