What's a Saddle Point?

Smythe

I'm working through a partial differentiation problem, to which I have the answer.

The function being 4x^2+4xy-y^3-2x+2 to which the stationary points must be obtained and classified.

At the end of the working out, when partial differentiation is conducted and a number of -32 is obtained, it states that because this is a negative number this is a "saddle point". Is this another term for a "point of inflection"?

How can they make this claim? I thought since the number obtained is negative it would therefore be a maximum turning point.

The second number obtained with the second partial differentiation is 32, and it states since this is a positive number this is a minimum turning point; which I understand.

Thank you.

**Timbuk2**

There is a saddle point at (a,b) if M(a, b) < 0, evaluated as the above.


Here's a graph of your function, courtesy of Wolfram Alpha, the red dot indicates the saddle point - it is a critical point, but it is neither a local maximum or a local minimum - which makes sense from looking at the graph, as it curves up in one direction, but curves down in another, like a mountain pass (or a horse saddle, hence the naming!).

See this for more info on using the second derivative test: http://mathworld.wolfram.com/SecondDerivativeTest.html

Smythe

Thanks very much for that Timbuk2

**Timbuk2**

No problem. That image doesn't seem to be displaying anymore (it was working earlier)
to see it, go to http://www.wolframalpha.com/
and type in saddle points 4x^2+4xy-y^3-2x+2
and press enter!

Tau

Mathematicians are generally good at picking sensible names for things - saddle point is one of these, but is a bit out-dated. A more modern name would be "pringle point".

The very middle of a pringle is flat, but its not the top of a peak, or the bottom of a hollow - its curving up one way and down the other.