Tool to determine the minimum value of a function: the minimal value that can take a function. It is a global minimum and not a local minimum.

Minimum of a Function - dCode

Tag(s) : Mathematics

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

Tool to determine the minimum value of a function: the minimal value that can take a function. It is a global minimum and not a local minimum.

For any function $ f $ defined on an interval $ I $ and $ m $ a real of this interval, if $ f(x) <= f(m) $ on the interval $ I $ then $ f $ reaches its minimum in $ x=m $ over $ I $. In that case, $ f(m) $ is the minimum value of the function, reached when $ x=m $.

The **minimum of a function** is always defined with an interval (which may be the domain of definition of the function).

Finding the **minimum of a function** $ f $, is equivalent to calculate $ f(m) $. To find $ m $, use the derivative of the function. The minimum value of a function is found when its derivative is null and changes of sign, from negative to positive.

__Example:__ $ f(x) = x^2 $ defined over $ \mathbb{R} $, its derivative is $ f'(x) = 2x $, that is equal to zero in $ x = 0 $ because $ f'(x) = 0 \iff 2x = 0 \iff x=0 $. The derivative goes from negative to positive in $ x = 0 $ so the function has a minimum in $ x=0 $, $ f(x=0) = 0 $ and $ f(x) >= 0 $ over $ \mathbb{R} $.

Add one or more conditions indicating the interval constraints for each variable.

__Example:__ Find the minimum of $ \sin{x} $ for $ 0 < x < \pi $

Indicate several equations with the operator logical AND && to separate the equations

An extremum is an extreme value of a function, this value can be maximum (the maximum value of the function) or minimum (the minimum value of the function).

dCode retains ownership of the online 'Minimum of a Function' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Minimum of a Function download for offline use on PC, tablet, iPhone or Android !

Please, check our community Discord for help requests!

minimum,function,derivative,calculation,maximum,extremum

Source : https://www.dcode.fr/minimum-function

© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲