local maximum

Someone will have to make a Mathematics-related former nodeshell Metanode but it isn't going to be me.


In Differential calculus, a local maximum is a point in a function of one or more variables which, as you might guess, is greater than all other values in its vicinity.

You can detect a local maximum by examining its first and second derivatives.

For one variable, we can say y = F(x). Then we examine every value of X where F'(x)=0. For each of these values x_i, if F"(x_i) also exists for that value, the point (x_i, F(x_i)) is a local maximum if F"(x_i) < 0. (If F"(x_i) > 0, it's a local minimum).

For example, let F(x)=3x³-x. Since F'(x)=9x²-1, there are two values where F'(x)=0 : +¹/₃ and -¹/₃. Since F"(x) = 18x, F"(+¹/₃)=6 means that +¹/₃ is a local minimum, and F"(-¹/₃)=-6 means that -¹/₃ is a local maximum.

For multiple variables, we find all points (x₁, x₂, ... x_n) where @F/@x₁ = @F/@x₂ = ... = @F/@x_n = 0. For each of these points, we calculate @²F/@x₁², @²F/@x₂², ... ,@²F/@x_n². If ALL of these second derivatives for a particular point are < 0, that point is a local maximum. If they are all > 0, it is a local minimum. If some are < 0, and some are > 0, it is a saddle point.