Functions can have points {optimum, calculus} {optima} where they are greatest or least. Functions can have highest points {relative maximum} in intervals. Functions can have lowest points {relative minimum} in intervals.
derivative
At maximum or minimum, derivative equals zero or has no definition. If first derivative changes sign, point is relative maximum or minimum.
Two-variable function maximum and minima are at points where both partial derivatives are zero. Maximum is if second partial derivative with respect to variable is less than zero: D^2f(x,y) / Dx < 0, where D^2 is second partial derivative, D is partial derivative, x and y are variables, and f is function. Minimum is if second partial derivative with respect to variable is greater than zero: D^2f(x,y) / Dx > 0.
Product of second partial derivatives with respect to each variable minus product of second derivatives with respect to each variable must be greater than zero: (D^2f(x,y) / Dx) * (D^2f(x,y) / Dy) - (d^2f(x,y) / dx^2) * (d^2f(x,y) / dy^2) > 0, where D^2 is second partial derivative, D is partial derivative, d^2 is second derivative, d is derivative, x and y are variables, and f is function.
Mathematical Sciences>Calculus>Differentiation>Optima
3-Calculus-Differentiation-Optima
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Date Modified: 2022.0224