# Min

Min[x1,x2,]

yields the numerically smallest of the xi.

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

yields the smallest element of any of the lists.

# Examples

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## Basic Examples(3)

Minimum of two numbers:

Minimum 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 minimum of all elements of a matrix:

The minima of all rows:

The minima of all columns:

For Interval objects Min gives the minimum element in all intervals:

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

### Specific Values(5)

Values of Min at fixed points:

Values at infinity:

Evaluate symbolically:

Solve equations and inequalities:

Find a value of x for which Min[{Sin[x],Cos[x]}]1/2:

### Visualization(3)

Plot the Min of several functions:

Plot Min in three dimensions:

Plot Min of three functions in three dimensions:

### Function Properties(9)

Min is only defined for real-valued inputs:

The range of Min is all real numbers:

Min effectively flattens out all lists:

Basic symbolic simplification is done automatically:

Additional simplification can be done using Simplify:

Multi-argument Min is generally not an analytic function:

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

Min can have any monotonicity depending on its arguments:

is not surjective:

Min 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(4)

Use in bounds of iterator variables:

Cumulative minima:

Find the lowest 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, Min returns Infinity:

Min is Flat and Orderless:

Use PiecewiseExpand to express Min and Max as explicit cases:

Use FullSimplify to simplify Min expressions:

Minimize a function containing Min:

Min can be differentiated:

## Possible Issues(2)

Min flattens lists, rather than being Listable:

Oneargument form evaluates for any argument:

## Neat Examples(2)

Two-dimensional sublevel sets:

Three-dimensional sublevel sets:

Wolfram Research (1988), Min, Wolfram Language function, https://reference.wolfram.com/language/ref/Min.html (updated 2021).

#### Text

Wolfram Research (1988), Min, Wolfram Language function, https://reference.wolfram.com/language/ref/Min.html (updated 2021).

#### CMS

Wolfram Language. 1988. "Min." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Min.html.

#### APA

Wolfram Language. (1988). Min. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Min.html

#### BibTeX

@misc{reference.wolfram_2024_min, author="Wolfram Research", title="{Min}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Min.html}", note=[Accessed: 18-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_min, organization={Wolfram Research}, title={Min}, year={2021}, url={https://reference.wolfram.com/language/ref/Min.html}, note=[Accessed: 18-July-2024 ]}