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Point of Inflexion
By Zhidong Leong on Monday, July 01, 2002 - 06:09 am:

What is the gradient of a point of inflexion of a curve? Eg: y = x3 How do I measure it?
Also, is it that all cubic equation has a pt of inflexion?

By Sam Davies on Monday, July 01, 2002 - 08:13 am:

A point of inflexion is defined as a point on a curve where the concavity changes. To find these points we look at the second derivative of the equation of the curve, f(x). At the point of inflexion f''(x) (i.e. d2y/dx2) = 0 and is changing sign. i.e. f'''(x) ¹0 or if x0 is the x value of the point of inflextion, then f''(x0 + a) < 0 and f''(x0 - a) > 0 (a is a constant), or vice versa. Therefore when you find the points of inflexion you naturally have the x values (x0). So the gradient at the point of inflexion here is just f'(x0), you either have the equation for f'(x) or can find it by integrating f''(x).

Have to dash to work now - unless somebody else does the other bit, i'll give it a stab later.

Cheers

sam

By Jamie Whale on Monday, July 01, 2002 - 10:56 pm:

Most cubic equations don't have one single point of inflection, a typical cubic equation with positive x3 in it (and various other quantities of x2 and x and c) would have a maximum point, then a minimum point.

By Brad Rodgers on Tuesday, July 02, 2002 - 01:24 am:

I would think all cubic equations have one and only one point of inflection: y = ax3 + bx2 + cx + d implies d2y/dx2 = 6ax + 2b, which changes from negative to positive, or vice-versa at it's zero, which occurs at x = -b/(3a).

Jamie's right about the local maximum and local minimum; in general a cubic has one local max and one local min. In fact either

a) the cubic has no local max. or min, but a point of inflection,

or

b) the cubic has a local max. and min. and also a point of inflection, all distinct

Brad

By David Loeffler on Tuesday, July 02, 2002 - 09:38 am:

Remember that a point of inflexion need not be a stationary point. For example, f(x) = x3-3x has an inflexion point at 0 (draw it) and maxima and minima at ±1.

David

By Zhidong Leong on Tuesday, July 02, 2002 - 12:42 pm:

So what exactly is a point of inflexion?

By George Walker on Tuesday, July 02, 2002 - 01:58 pm:

basically where the curve is 'straight'
like y=sinx at x=0 the line is changing concavity, with gradient 1.

By David Loeffler on Tuesday, July 02, 2002 - 02:14 pm:

Literally "inflexion" = "no curvature", neither concave nor convex.

By Brad Rodgers on Tuesday, July 02, 2002 - 08:16 pm:

The point also must mark a transition from concave up to concave down (or from concave to convex, depending upon which terminology you use). So the vertex of a quadratic is not a point of inflection.

I think that's already been said, but I've said it again anyways.

Brad

By David Loeffler on Wednesday, July 03, 2002 - 01:38 pm:

Hmm. I would say f(x) = x4 has a point of inflexion at the origin despite being convex everywhere.

David