# Max Max[x1,x2,]

yields the numerically largest of the xi.

Max[{x1,x2,},{y1,},]

yields the largest element of any of the lists.

# Details # Examples

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## Basic Examples(3)

Maximum of two numbers:

Maximum of a list:

Plot over a subset of the reals:

## Scope(27)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

The maximum of all elements of a matrix:

The maxima of all rows:

The maxima of all columns:

For Interval objects, Max gives the maximum element in all intervals:

For CenteredInterval objects, Max[Δ1,Δ2] gives an interval containing Max[a1,a2] for any aiΔi:

### Specific Values(5)

Values of Max at fixed points:

Values at infinity:

Evaluate symbolically:

Solve equations and inequalities:

Find a value of x for which Max[{Sin[x],Cos[x]}]1:

### Visualization(3)

Plot the Max of several functions:

Plot Max in three dimensions:

Plot Max of two functions in three dimensions:

### Function Properties(9)

Max is only defined for real-valued inputs:

The range of Max is all real numbers:

Max effectively flattens out all lists:

Basic symbolic simplification is done automatically:

Additional simplification can be done using Simplify:

Multi-argument Max is generally not an analytic function:

It will have singularities where the arguments cross, but it will be continuous:

Max can have any monotonicity depending on its arguments: is not surjective:

Max can have any sign depending on its arguments:

### Differentiation and Integration(5)

First derivative with respect to x:

Higher derivatives with respect to x:

Formula for the  derivative with respect to x:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integrals:

## Applications(5)

Use in bounds of iterator variables:

Cumulative maxima:

Find the highest point of a plotted curve:

Mean of the length ratio of a randomly broken stick:

Rfunction-based solid modeling:

## Properties & Relations(6)

With no arguments, Max returns :

Max is Flat and Orderless:

Use PiecewiseExpand to express Max and Min as explicit cases:

Use FullSimplify to simplify Max expressions:

Maximize a function containing Max:

Max can be differentiated:

## Possible Issues(2)

Max flattens lists, rather than being Listable:

The oneargument form evaluates for any argument:

## Neat Examples(2)

Two-dimensional sublevel sets:

Three-dimensional sublevel sets: