Saddle Point

 

Saddle Point

Week 13

Thanksgiving was yesterday and I ate turkey dinners at home with my family. Today several stores such as Target, Walmart, and Kmart are opening their doors hours during Black Friday. Hundreds shoppers will be able to take advantage of special deals available during this time.

As usual, I am thinking mathematically how can see this procurement. How to minimize the total procurement cost considering discount prices.

 

I can find the saddle point which is a point in the range of a function that is a stationary point but not a local extremum. So, I can find the saddle point for a one-dimensional function, for example:

f(x)=x^3,

 f^'(x) = 3x^2

f^('')(x) = 6x  

f^(''')(x) = 6.

This function has a saddle point at x₀=0 by the extremum test since:

f^('')(x₀)=0 and f^(''')(x₀)=6≠0.

In one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.