(Source: demonstrations.wolfram.com)

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F''(x) = 6x f''(0) = 0 Sgt; x = 0 is a point of inflectionPoints of inflection do not necessarily have a zero gradient, unlike maxima and minima which must. In the context of this graph, the inflection point occurs when the value of the line stops increasing relative to the vertical axis and the slope becomes zero.

Generally, this is the point at which the individual transitions from aerobic to anaerobic metabolism. For most individuals, the duration of time for which anaerobic activity can be sustained is limited and can be measured in minutes (as opposed to hours).

As far as I know, saddle points deal with three-dimensional surfaces ... what is your definition of a saddle point? Basically, it is just what skeeter said- an inflection point is a point on a two-dimensional graph, y= f(x), where the curvature changes from “concave upward” to “concave downward”.

But the graph at x = 2 is a corner which is not an inflection point Since of'$ and of''$ do not exist at that point, I wouldn't call it a saddle point. Check with your instructor to verify the definition he wants you to use.

In the case when the function is a parabola, a calculation shows that, for xed h, the area of ABP is constant even as the point a varies (see Figure 2). This geometric curiosity may well have been known to the ancient Greeks of course.

(Source: demonstrations.wolfram.com)

What about the converse question: for which functions am Area(a , h) independent of a ? The theorem we just proved provides an answer to that: only parabolas (and lines, in which case all the triangles are degenerate and the areas vanish).

Let f have a continuous non-zero second derivative on an open interval I. If, for all x in I, Area(x, h) depends only on h, then the graph of f is a parabola, and conversely.

The author wishes to thank the referee for several suggested improvements. In the study of surfaces in multivariate calculus, we notice some similarities be- tween saddle points on surfaces and inection points on curves.

In this note, we make a direct connection between the two concepts. Throughout, we assume that f (x, y) has continuous second partial derivatives in an open set in the plane, and that (a , b) is a critical point in that set (that is, f x (a , b) = f y (a , b) = 0).

The standard test for extreme uses the discriminant D = AC B 2 : f has a relative maximum at (a , b) if D > 0 and Let f be a function with continuous second partial derivatives in an open set U in the plane and let (a , b) be a saddle point in U.

(Source: math.stackexchange.com)

Then there exists a continue- out function y = g (x) with g (a) = b for which the projection on the x z-plane of the intersection of the surface z = f (x, y) and the cylindrical surface y = g (x) has an inection point at x = a. As an example, we consider the function f (x, y) = y 2 x 2, which has a saddle point at (0, 0) as shown in Figure 1.

In Figure 2 we show the intersection between the surfaces z = f (x, y) and y = g (x) . In Figure 3 we show this intersection from the point of view of the y -axis, and we can observe that the projection on the x z -plane has a point of inection at x = 0 (Figure 4).

Tion g (x) obtained in the proof satises the theorem but is not differentiable at x = a In Figure 5 we show the intersection between the surfaces z = y 2 x 2 and y = x x 2.

In Figure 6 we show this intersection from the point of view of the y -axis, and in Figure 7 we observe that the projection on the x z -plane has a point of inection at x = 0. We don’t know the answer, but we count on some readers to explore this question.

Sidney H. King (sidneykung@yahoo.com), Cupertino, CA 95014 We give an analytic proof of the fact that the conic sections are obtained by cutting a cone at various angles.

(Source: demonstrations.wolfram.com)

Our proof does not involve spheres or circles (see ), but primarily depends upon the cutting plane itself. Figure 1 shows a two-napped circular cone C which may be viewed as the result of rotating the line g (generator) about the xed line l (z -axis) while maintaining the same angle () between g and l.

Given a matrix of n × n size, the task is to find the saddle point of the matrix. A saddle point is an element of the matrix such that it is the minimum element in its row and maximum in its column.

A simple solution is to traverse all matrix elements one by one and check if the element is Saddle Point or not. An efficient solution is based on the below steps. Find the minimum element of the current row and store the column index of the minimum element.

Exercise : Can there be more than one SaddlePoints in a Matrix? This article is contributed by Sail Chiara(KILLER). If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org.

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