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 xi, if F"(xi) also exists for that value, the point (xi, F(xi)) is a local maximum if F"(xi) < 0. (If F"(xi) > 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, ... xn) where @F/@x1 = @F/@x2 = ... = @F/@xn = 0. For each of these points, we calculate @2F/@x12, @2F/@x22, ... ,@2F/@xn2. 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.

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