Search for a tool
Maximum of a Function

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

Results

Maximum of a Function -

Tag(s) : Functions

Share dCode and you

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!

Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best Maximum of a Function tool. Thank you.

# Maximum of a Function

## Maxima Calculator

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

### What is the definition of a maximum of a function?

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 maximum in $$x=m$$ over $$I$$. The value of the maximum is $$f(m)$$.

Example: $$f(x) = -x^2$$ defined over $$\mathbb{R}$$, the function reaches its maximum in $$x=0$$, $$f(x=0) = 0$$ and $$f(x) <= 0$$ over $$\mathbb{R}$$.

The maximum of a function is always defined with an interval, it can be local (between 2 values), or global : over the domain of definition of the function.

### How to calculate a maximum of a function?

The maximums of a function are detected when the derivative becomes null and changes its sign (passing through 0 from the positive side to the negative side).

Example: Calculate the maximum of the function $$f(x) = -x^2 + 1$$. This function has for derivative $$f'(x) = -2x$$ which is nullable in $$x = 0$$ as $$f'(x) = 0 \iff -2x = 0 \iff x = 0$$. An extremum is found in 0, its value is $$f(0) = 1$$. Calculations of limits $$\lim_{x\to0^-}{f'(x) = 0^+} \\ \lim_{x\to0^+}{f'(x) = 0^-}$$ show that the derivative change of sign from positive $$0^+$$ to negative $$0^-$$. Global extremun of the function is then $$1$$ for $$x=0$$.

dCode has also a minimum of a function calculator tool.

### What is an extremum?

An extremum is the name given to an extreme value of a function, a value that can be maximum (maximum of a function) or minimal (minimum of a function).

## Source code

dCode retains ownership of the source code of the script Maximum of a Function online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Maximum of a Function script for offline use on PC, iPhone or Android, ask for price quote on contact page !