Built-in Mathematica Symbol

FindMaximum

FindMaximum[f, {x, x₀}]
searches for a local maximum in f, starting from the point x=x₀.

FindMaximum[f, {{x, x₀}, {y, y₀}, ...}]
searches for a local maximum in a function of several variables.

FindMaximum[{f, cons}, {{x, x₀}, {y, y₀}, ...}]
searches for a local maximum subject to the constraints cons.

FindMaximum[{f, cons}, {x, y, ...}]
starts from a point within the region defined by the constraints.

FindMaximum returns a list of the form {f_max, {x->x_max}}, where f_max is the maximum value of f found, and x_max is the value of x for which it is found.

If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.

The constraints cons can contain equations, inequalities or logical combinations of these.

FindMaximum first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.

FindMaximum has attribute HoldAll, and effectively uses Block to localize variables.

FindMaximum[f, {x, x₀, x₁}] searches for a local maximum in f using x₀ and x₁ as the first two values of x, avoiding the use of derivatives.

FindMaximum[f, {x, x₀, x_min, x_max}] searches for a local maximum, stopping the search if x ever gets outside the range x_min to x_max.

Except when f and cons are both linear, the results found by FindMaximum may correspond only to local, but not global, maxima.

For linear f and cons, xIntegers can be used to specify that a variable can take on only integer values.

AccuracyGoal	Automatic	the accuracy sought
EvaluationMonitor	None	expression to evaluate whenever f is evaluated
Gradient	Automatic	the list of gradient functions {D[f, x], D[f, y], ...}
MaxIterations	Automatic	maximum number of iterations to use
Method	Automatic	method to use
PrecisionGoal	Automatic	the precision sought
StepMonitor	None	expression to evaluate whenever a step is taken
WorkingPrecision	MachinePrecision	the precision used in internal computations

The settings for AccuracyGoal and PrecisionGoal specify the number of digits to seek in both the value of the position of the maximum, and the value of the function at the maximum.

FindMaximum continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.

Possible settings for Method include "ConjugateGradient", "Gradient", "LevenbergMarquardt", "Newton" and "QuasiNewton", with the default being Automatic.