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)=3x

^{3}-x. Since F'(x)=9x

^{2}-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 (x

_{1}, x

_{2}, ... x

_{n}) where @F/@x

_{1} = @F/@x

_{2} = ... = @F/@x

_{n} = 0. For each of these points, we calculate @

^{2}F/@x

_{1}^{2}, @

^{2}F/@x

_{2}^{2}, ... ,@

^{2}F/@x

_{n}^{2}. 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.