Relative minima {minimum, curve} have first derivative zero and second derivative greater than zero.
Relative maxima {maximum, curve} have first derivative zero and second derivative less than zero.
Functions {concave curve} can have points where second derivative is greater than zero {concave upward, concave curve} {convex downward, concave curve} and tangent is below curve. Functions can have points where second derivative is less than zero {concave downward, concave curve} {convex upward, concave curve} and tangent is above curve.
Functions {convex curve} can have points where second derivative is greater than zero {concave upward, convex curve} {convex downward, convex curve} and tangent is below curve. Functions can have points where second derivative is less than zero {concave downward, convex curve} {convex upward, convex curve} and tangent is above curve.
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.
Function can have points {inflection point}| where second derivative equals zero, tangent intersects curve, and curve has zero curvature. At inflection point, curve changes from concave to convex, or vice versa.
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Date Modified: 2022.0225