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 @
2F/@x
12, @
2F/@x
22, ... ,@
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.